MEASUREMENT OF THE COMPLEX DYNAMIC RIGIDITY OF RECENT MARINE SEDIMENTS Gregory Allen Engel Library Naval Postgraduate School Monterey, California 93940 POSTGRADUATE SCHOOL Monterey, California MEASUREMENT OF OF THE COMPLEX DYNAMIC RECENT MARINE SEDIMENTS RIGIDITY by Gregory Aller i Engel Th ssis Co- -Advisors : 0. R. B. S. Wilson Andrews December 1972 ApfVioved Ion. pubtic A.eXeo6e; dUtAAbution antinuXtd. Measurement of the Complex Dynamic Rigidity of Recent Marine Sediments by Gregory Allen Engel Lieutenant Junior Grade, United States Navy B.S., United States Naval Academy, 1971 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL December 1972 Th* Library Naval Postgraduate School Monterey, California 93940 ABSTRACT The dynamic rigidity of 17 samples of continental terrace clayey- silt sediments has been measured in the laboratory using a viscoelast- ometer in the frequency range of 7 to 60 kHz. The method involves the propagation of torsional waves on a rod and measuring the effects of shear loading imparted to the rod when imbedded in a sediment. Values of the real component of rigidity range from 1.6 X 10" dynes/cm^ to 2.1 X 10' dynes/cm . Values of the imaginary component of rigidity range from 2.0 X 10^ dynes/cm^ to 4.1 X 10^ dynes/cm^. No clear-cut depen- dence of rigidity upon frequency is observed. Both real and imaginary components of rigidity are analyzed by plotting the data as a function of various other mass-physical properties, including: density, porosity, compressional wave speed, sand-silt-clay percentages, vane shear strength, and the product of density and sound speed squared. These analyses substantiate research done by other workers indicating that both real and imaginary components of rigidity exhibit trends with some of the mass-physical properties. TABLE OF CONTENTS I. INTRODUCTION 8 II. THEORY AND METHOD OF MEASUREMENT 10 A. GENERAL 10 B. RESONANCE METHOD 12 III. EQUIPMENT AND PROCEDURES 14 A. EQUIPMENT 14 B. PROCEDURES 16 C. SEDIMENT PREPARATION 16 IV. LIMITATIONS ON RESULTS 20 V. RESULTS AND DISCUSSION 24 VI. CONCLUSIONS AND RECOMMENDATIONS 30 COMPUTER PROGRAMS 73 BIBLIOGRAPHY 77 INITIAL DISTRIBUTION LIST 79 FORM DD 1473 81 LIST OF TABLES I. Station Location, Water Depth, and Bottom Water Temperature 31 II. Textural Analysis of 6-inch Core Sections 32 III. Textural Analysis of 14-inch Core Sections ( Int erpo lated) 33 IV. Mass-Physical Properties of 6-inch Core Sections 34 V. Mass-Physical Properties of 14-inch Core Sections (Interpolated) 35 VI. Mass-Physical Properties of 14-inch Core Sections 36 VII. Acoustic Properties 37 VIII. Values of Viscoelastic Calibration Coefficients Kx and K2 38 IX. Standard Deviation and Variation of Dynamic Rigidity Parameters 39 LIST OF FIGURES 1. Coordinates for Shear Wave Propagation, Equivalent Electrical Circuit of Rod-Transducer System 41 2. Cut-Away View of Viscoelastometer and Shear Transducer Polarities 42 3. Block Diagram of Equipment for Resonant Method 43 4. Coring Station Locations 44 5. Sectioning of Cores 45 6. Plot of Sand/Silt/Clay Ratios for Each Core Section Analyzed 46 7. Resonant Frequency as a Function of Temperature Probe 1, Mode 1 47 8. Resonant Frequency as a Function of Temperature Probe 1, Mode 2 48 9. Resonant Frequency as a Function of Temperature Probe 1, Mode 3 49 10. Resonant Frequency as a Function of Temperature Probe 1, Mode 4 50 11. Resonant Frequency as a Function of Temperature Probe 2, Mode 1 51 12. Resonant Frequency as a Function of Temperature Probe 2, Mode 2 52 13. Resonant Frequency as a Function of Temperature Probe 2, Mode 3 53 14. Resonant Frequency as a Function of Temperature Probe 4, Mode 1 54 15. Resonant Frequency as a Function of Temperature Probe 4, Mode 2 55 16. Resonant Frequency as a Function of Temperature Probe 4, Mode 3 56 17. Gj as a Function of Density 57 18. G^ as a Function of Porosity 58 19. G^ as a Function of Sound Speed 59 20. G\ as a Function of Percent Sand : 60 21. G^ as a Function of Percent Silt 61 22. Gj as a Function of Percent Clay 62 23. G^ as a Function of Vane Shear Strength 63 24. Gi as a Function of Density and Sound Speed Squared 64 25. G2 as a Function of Density 65 26. G2 as a Function of Porosity 66 27. G2 as a Function of Sound Speed 67 28. G2 as a Function of Percent Sand : 68 29. G2 as a Function of Percent Silt 69 30. G2 as a Function of Percent Clay 70 31. G2 as a Function of Vane Shear Strength 71 32. G2 as a Function of the Product of Density and Sound Speed Squared 72 Acknowledgements The author wishes to express his gratitude to Professor 0. B. Wilson and Professor R. S. Andrews of the Naval Postgraduate School for their guidance and aid in the research described in this report. The author also wishes to thank Lieutenant Commander Robert J. Cepek, USN, and Mr. Ken Smith for their support and the associated research of this report. Additionally, the author gratefully ac- knowledges the assistance of Mr. Larry Leopold of San Jose State College and his students in coring operations at sea. Use of the USNS BARTLETT (T-AG0R-13) was essential in obtain- ing core samples for research. The project received support through contract with the Office of Naval Research, Ocean Science and Technology Division. I. INTRODUCTION Because of the growing importance of sound reflections from the ocean floor during recent years, an increasing amount of study has been direc- ted toward marine sediments with the goal of improving the accuracy of predicting the acoustic reflection characteristics of the ocean floor, taking into account variations in sediment constituency and mass-physical properties. Various acoustic models of the ocean floor have been used in these efforts. Most assume a multilayered liquid model, the simplest of which is the two-fluid model where the sediment is considered to be a layer of fluid with a density and sound speed different from that of the overlying sea water. Such models are often successful because of low rigidities and shear-wave velocities in surficial sediments [8] . Somewhat more so- phisticated models include the effects of inhomogeneities in the bottom materials and the effects of a bottom which has a shear elasticity. A still more realistic model for sediments is that of a viscoelastic solid. Several studies have shown that although elastic model equations ade- quately define the velocities of compressional and shear waves, they do not, of course, account for absorption, which is also important [9]. For this reason a viscoelastic model involving complex Lame constants and including the generation of shear waves upon reflection and ab- sorption of the waves is favored [9] . An additional problem is the dif- ficulty of evaluating the Lame constants for various oceanic areas and soft sediment types. Evaluation of the constants is known by only three direct methods: (1) the Stonely wave technique [10], which is limited to low frequencies; (2) use of torsional wave vibration [7]; or (3) a torsional wave viscoelastometer [11, 5, 2]. 8 Research conducted by Hutchins [11] at the Naval Postgraduate School demonstrated the feasibility of measuring the complex shear modulus or dynamic rigidity of simulated ocean sediments. The method involves mea- suring the effects of shear waves generated in the sediment due to the propagation of torsional waves along the length of a cylindrical rod im- mersed in the sediment. Hutchins worked with kaolinite-water mixtures in the narrow frequency range of 38.3 to 38.9 KHz. Cohen [5] used an improved model viscoelastometer and extended the frequency range to 5.8 to 38 KHz. Using saturated samples of kaolinite and bentonite clays, Cohen demonstrated that these soft, undispersed, simulated sediments do have measurable shear moduli which are independent of frequency. Bieda [2] developed a totally immersible viscoelastometer and used it to deter- mine the complex rigidity of actual ocean sediments obtained from the continental terrace. Bieda analyzed the sediments for mass-physical properties and demonstrated that both real and imaginary components of rigidity exhibit trends with some of those properties. The purpose of the research here reported was to test and develop a newer model viscoelastometer, possibly suitable for in situ measurement, and to apply it to the measurement of dynamic rigidity of real ocean sediments. The following sections describe the theory of measurement, the equipment and procedures, sample collection and preparation, the limitations of the method, and results of the various rigidity determi- nations. A discussion of results and recommendations for future research are then presented. II. THEORY AND METHOD OF MEASUREMENT A. GENERAL The determination of the sediment's shear modulus and viscosity is made from measurements of the reaction to shear waves which radiate into the sediment from the surface of a cylindrical metal rod which is oscil- lating in torsion. This torsional motion, generated by a piezoelectric crystal transducer mounted within the metal rod, is at a frequency for resonance of the waves propagating in the rod. Although the shear waves in the sediment are usually too highly attenuated to allow investigation of their propagation, the reaction to this radiation is detected by elec- trical measurements on the piezoelectric crystal. The geometry of the torsional oscillation and the resulting shear waves is indicated by the sketch in Figure 1. The rod axis about which simple harmonic torsional oscillations take place is parallel to the y-direction. At the circumference of the rod, displacements are along the circumference, the x-direction. Shear waves in the surrounding medium, generated by this motion, propagate radially outward, along the z-direction. Following Mason's development [13] and McSkimin's hypothesis [15], if it can be assumed that shear waves propagating radially outward into a fluid in contact with the rod have a wave-length which is very small compared to the radius of curvature of the rod, and that the amplitude of these waves is rapidly attenuated, then the waves can be treated essentially as plane waves propagating in a fluid having a shear vis- cosity coefficient, n. Plane shear waves of the form UY = Une-,zeiCut-kz) CI) 10 propagating into an infinite medium are assumed. For a Newtonian fluid, of = k =-J7fP (1A) / n where f = frequency, p = density and n = shear viscosity coefficient. U0 is the initial amplitude. If the sediment were solid, its shear modulus would be given by: G = t/s (2) where t is shear stress, and S is shear strain. In a liquid sediment the shear viscosity coefficient would be n = x/S, where S = 9S/3t, and is thus the shear velocity. Assuming a viscoelastic sediment, both G and n are complex [14]. Gc = G1 + iG2 (3) nc = r\x - in2 (4) G-p = 0 in a perfectly elastic solid; n2 = 0 in a Newtonian fluid. For plane, simple, harmonic waves of angular frequency co, nc = T/iwS = -iGc/a) (5) Hence, G^ = wn2, and G2 = wrii, where to is the angular frequency. The specific acoustic load impedance of the sediment on the rod, Z, is given by Z = ncUx and for plane shear waves is given by Z'-Y^Psed^1 + i) = R + iX (6) where R = specific acoustic resistance of the sediment 11 X = specific acoustic reactance of the sediment p , = density of the sediment sed 3 Substituting nc in its complex form and separating real and imagi- nary parts yields m = 2RX/Wpsedj n2 = (R2 - X2)/a)Psed. (7) G1 = (R2 - X2)/Psed) G2 = 2RX/Psedf (g) Determination of shear moduli and viscosities then reduces to measuring R and X. B. RESONANCE METHOD In this method, based on work by Mason [13] , the torsional trans- ducer-rod combination is driven in a continuous wave mode, exciting standing waves on the probe. The rod-transducer system is represented approximately by an equivalent electrical circuit consisting of the capacitance, C, in parallel with a series RLC circuit which arises from the motions of the system. When excited at the frequency of mechanical resonance the electrical conductance, Gp, is a maximum. At each reso- nance mode the input electrical conductance is measured first with the probe suspended in a vacuum, then with the probe imbedded in the medium under study. This allows calculation of the change in equivalent par- allel electrical resistance at resonance, ARp. The change in the reso- nance frequency for each mode, Af, is also recorded. Mason has shown that [13] : ARE = K:RT and Af = k2Xt where K, and K2 are constants which are a function of fixed physical parameters. These constants are determined by making measurements with 12 the probe suspended in Newtonian fluids of known viscosity. In a New- tonian fluid, n? = 0; therefore R-p = XT. R„ and X„ are the components of the total impedance presented to the rod by the sediment. After substitution, K, and K2 reduce to Kl = GFl " Gvac 1"VPF£ * fF£ t^1 WPF£ * fVl V F£ where G = electrical conductance 1 = length of viscoelastometer submerged p = density f = frequency nj = viscosity in poise (9) fvac " fFl K2 = ; /n r==i do) 13 III. EQUIPMENT AND PROCEDURES A. EQUIPMENT The driving transducer is a hollow barium titanate ceramic cylinder 0.5 inches in diameter and 1.5 inches in length. It is constructed from an axially-polarized cylinder which is cut into two parts along the lon- gitudinal axis and rejoined following inversion of one half-section and insertion of two electrode grids (Figure 2) . This produces a shear-type oscillation in the ceramic when electrically excited. A description of this method can be found in Mason [14] . Of the three laboratory viscoelastometers utilized, two were stain- less steel and one was constructed of a constant modulus alloy, Nl-Span- C. As in earlier probes used by Bieda the transducer is located at the center giving a maximum point of stress for all resonant modes. In con- trast to Bieda's probes, however, the newer model viscoelastometers have the transducers completely within the viscoelastometer body (Figure 2). Stainless steel was used because it does not corrode when immersed in sediment for long periods. Nl-Span-C was being tested for its more stable characteristics. Each rod is 20.32 cm long, 1.78 cm in diameter, and cut at mid-section to allow insertion of the ceramic oscillator. Both sections of the rod are polished. The Nl-Span-C rods were heat- treated to achieve constant modulus before assembly. The transducer is glued to each half of the viscoelastometer to join the rod. Small holes drilled in the internal perimeter of the rod allow the glue to penetrate and help to ensure proper bonding. A 5-mil thick nylon mesh gasket was used in the glue joint to control glue joint thickness. The electrical leads to the transducer are carried out through small holes at the joint 14 between the two halves of the metal rod. After the bonding glue had cured, the joining seam and the electrical connections were given a thin coat of epoxy to prevent leakage to the interior of the probe and to provide electrical insulation. At the top of the viscoelastometer a small hole was drilled. A few strands of stainless steel thread were then inserted, being held in place with low temperature solder. To the other end of the thread a small pinch clip was soldered for support purposes. On the exterior of each a probe near the seam a small coating of varnish was applied and a number scribed on each viscoelastometer. Probe 1 and Probe 2 are stainless steel. Probe 4 is the Nl-Span-C alloy. The vacuum calibrating was done within copper canisters with an in- side diameter of 6.3 cm. "0" rings prevented leakage at the metal to metal junctions of the canister. The vacuum canister containing the probe was then placed in a thermally controlled water bath at 20.0°C. Water temperature was monitored with a digital thermometer. To deter- mine the temperature dependence of the resonant frequencies of the probes in vacuum the water bath temperature was raised to 25°C. A sufficient period was allowed for the equipment to stabilize and the resonant fre- quencies of each mode were recorded. A similar process was again fol- lowed for 20°C, 15°C, 10°C and 5°C. The results of these studies are shown in Figures 7-16. The resonance method equipment configuration is shown in Figure 3. The sinusoidal signal generated by the oscillator is connected to the counter and the admittance meter in parallel. The five-decade counter was set for either a 10-sec or 1-sec gate in order to provide a preci- sion in the determination of the resonant frequency consistent with that with which the settings could be made. 15 Compressional wave speed in the sediment cores was measured with a sediment velocimeter obtained from the Pacific Support Group of the Naval Oceanographic Office. The velocimeter uses a two transducer meth- od to measure the time delay for sound transmission through the diameter of the core samples. Computation of the velocity in the sediment is ac- complished by comparing the time delay through the sediment filled core liner to the time delay in an identical core filled with distilled water at the same temperature. The equation is: Csed = Cw ( x _ cwA t } d where Cseci is the sediment compressional wave speed, Cw is the sound speed in distilled water, At is the comparable time delay, and d is the inside core diameter. B. PROCEDURES At resonance the viscoelastometer is driven in a standing mode and resonance frequencies as well as input electrical conductances at reso- nance are recorded. Measurements made in a vacuum, and later in Newton- ian fluids, permit calculation of K\ and K2 of the probes (Program KCALC) . A Brookfield Viscometer is utilized to obtain values of the shear vis- cosity coefficient for the calibrating fluid. The K values are obtained for each mode of vibration for each probe in the various calibrating fluids. The K values from the various calibrating fluids for each mode are then averaged for each probe. The final averaged values of K are listed in Table VIII. C. SEDIMENT PREPARATION The supply of sediment cores were obtained from operations conducted aboard USNS BARTLETT (T-AGOR-13) on 11-13 May 1972 in the area illustrated 16 in Figure 4. The area was chosen so as to insure that samples would contain minimal amounts of sand-sized particles. The area is on the Monterey Submarine Fan where previous coring and dredging has yielded samples of a low sand content [17] . Minimization of sand content is necessary as particles in this size range are comparable to the wave- length of the shear waves at the frequencies produced by the visco- elastometer. This invalidates the assumption of homogeneity. The samples retrieved were gravity cores. Upon being brought aboard the cores were cut into alternate 6-inch and 14-inch sections using the heated element method. This technique utilizes a soldering gun with a modified tip and positioning rings to cut the core liner only. A thin wire was passed through the sediment to separate the sections. As soon as possible the 6-inch sections were subjected to vane shear strength measurements in the ship's dry lab using a Wykeham-Farrance Vane Shear Machine with modifications. Output graphs of the torque vs. rotation at sediment failure were recorded. These vane shear strengths were then correlated with the values of dynamic rigidity obtained from viscoelas- tic measurements. A more detailed description of cutting, coring and vane shear procedures is given by Cepek [4] . The 14-inch core sections were capped and stored upright in the ship's wet lab refrigerator until they could be transferred to the Sea Floor Acoustics Laboratory at- the Naval Postgraduate School. Once trans- ferred they were placed in a salt water bath under refrigeration at 5°C until at such time they could be used for the viscoelastic study. A salt water bath was chosen to prevent dessication of the sediment through the core liner. 17 The 6-inch cores were tested -for determination of porosity. Por- osity is the percentage of voids per unit volume. Porosity of the 6-inch cores was calculated by Cepek [4] and the porosity of the 14-inch sections was determined from interpolation (Table V) . Laboratory measurements of wet density and water content were con- ducted simultaneously using prescribed procedures [12] and calculated by Cepek. Again the values of the wet density and water content of the 14- inch core sections were obtained from interpolation of the 6-inch core section values (Table V) . All sediment samples from the BARTLETT cruise were subjected to grain size analysis. The sieve analysis method was performed on the coarse fraction of each sample (<40) , and the pipette method was used for each sample after the coarse fraction had been removed. Again, results and description of the analysis are given by Cepek. For the 14-inch core sections each sediment sample was inspected upon probe removal. Coarse particles were never found in the vicinity of viscoelastic penetration. To prepare the 14-inch core sections for insertion of the viscoelas- tic probe, the sediment samples were removed from their 5°C salt water bath and immersed in a 20°C bath. After temperature stabilization was attained compressional wave speed measurements were made. The sediment core was then uncapped and a hollow coring tube of 1.75 cm diameter was slowly pushed vertically through the center of the sediment core. This provided a bore hole of a slightly smaller diameter than the viscoelas- tometer into which the viscoelastometer could be pressed with a minimum of sediment disturbance. No support for the probe was required as the sediment was sufficiently strong to maintain the viscoelastometer in position. With the electrical leads in position the core was then 18 recapped and taped to prevent moisture loss. The electrical conductivity of the sediment was sufficient to provide an electrical ground connection for the probe. / The recapped core section was then placed in a thermally controlled container and electrical connections were made. Between eight and ten hours were required before stable and consistent measurements could be taken. To extract the probe from the sediment a cork plunger was utilized. The probe and sediment were then removed and the sediment inspected for sand content in the vicinity of the probe. 19 IV. LIMITATIONS ON RESULTS One goal of the development of this viscoelastometer in these studies was to achieve values for K^ and K2 which were frequency independent. Complete independence was not accomplished; however, the stability was much greater than in previous studies with earlier model viscoelastometers. Various viscosity oils, ranging from 10 to 300 poise, were used with the higher viscosities giving better results. It was found that the visco- elastometers were unable to distinguish the lighter viscosity fluids from non-loaded conditions accurately and these results were neglected. A narrow range of higher viscosity fluids will give much better results for consistent values of K^ and K2. Average values of the constants were used to compute dynamic rigidities (Table V) . Two critical assumptions of theory development are that the wave- length of the shear wave in the medium is much smaller than the radius of curvature of the probe and that the amplitude decays to a small value in an insignificant distance compared to the radius of the rod. The shear wave speed, Cs, is calculated using the formula: cs = -VgT^ where G is the absolute value of the complex rigidity and p is the sed- iment density. The wavelength of the shear wave is then: X = Cs/f where f is the frequency. The wavelength in the stiff er sediments at the lowest mode is about .15 cm while the radius of curvature of the rod is about 0.9 cm. This assumption is therefore assumed valid at all frequencies of interest. 20 The absorption coefficient , °< (Equations 1 and 1A) , for a typical case is about 15 cm" . This implies that the shear wave amplitude is reduced to 1/e of its initial value in about 0.07 cm, again, signifi- cantly smaller than the radius of the rod. Another important assumption for linearizing the wave equation is that the amplitudes of motion are essentially infinitesimal. To deter- mine the validity of this assumption, the displacement of the visco- elastometer was optically measured at resonance through a high-powered microscope. In an unloaded state (air) and driven at 50 volts, the max- imum observed peak to peak displacement amplitude was about one micron at the lowest mode. Imbedded in sediment while resonating under normal conditions (driven at 3 volts) the amplitude will be considerably less than 0.06 microns. Thus the assumption of a small amplitude holds true. Since no sand-sized particles were found in the vicinity of the viscoelastometers during sediment analysis, no anomalous behavior is expected for this reason. There were other problems which affected the accuracy of the results. One was the mechanical degradation of the viscoelastometers in service. The following case is one example. Early in the test program, after completing a sequence of measurements in a number of cores, viscoelas- tometer No. 1 was rechecked in the vacuum chamber. Corrosion of the support hook had required that this be replaced, which involved heating one end of the rod to the temperature for melting soft solder. It was observed that the resonance frequencies had shifted significantly, as much as 95 Hz for one mode, since the first calibration tests. Although the heating process was done carefully, this may have changed the elastic properties of the rod or have disturbed the glue joints. In any case, this casts some uncertainties on the results gained with this probe. 21 Another problem involved leakage of water into the transducer com- partment of the torsional probes. This appeared to be due to repeated immersions and washings which degraded the epoxy-resin seal at the press fit joint between the two halves and at the electrical lead-in wires. The disturbance to the measurements caused by the resulting electrical leakage was usually severe enough that this trouble was detected almost immediately. In most cases, the trouble was corrected by scraping away the epoxy-resin seal, drying by pumping in a vacuum and resealing. Additional reasons for uncertainties in the obtained results are the structural damage caused by the core collection method and by possible core dessication during storage. The cores were gravity cores and a con- siderable amount of sediment disturbance is assumed. Analysis utilizing Probe 1 was commenced shortly after the cores were retrieved. Analysis utilizing Probe 2, however was not carried out for another five months. Even though the cores were kept in a 5°C salt water bath, various amounts of dessication and compaction were noted to have occurred. Through qual- itative handling the sediments in this second group were found to be firmer, which one would expect to be associated with increased values of rigidity. This also implies that the textural sediment analysis which was undertaken when the cores were fresh will not strictly apply to the measurements from Probe 2. To determine errors due to uncertainties in the measurements, the standard deviation of each input parameter was determined (Table IX). These standard deviations were then used to recalculate the values of dynamic rigidity from which the percentage variation could be computed. This resulted in a maximum possible relative error of about 50% in Gi, and about 72% in C>2- Such a large error is attributable mainly to a 22 sensitivity to values of the resonant frequency and conductance. An uncertainty of 0.67 Hz in the probe's resonant frequency in a sediment results in a 37% uncertainty in G-,. A deviation of this amount can be caused by temperature, variation, instrument error, measurement error or a combination of them. 23 V. RESULTS AND DISCUSSION The original intent of the current viscoelastometer production was to develop an in situ probe. Temperature dependence, fragile physical composition, and variability of calibration constants indicate this model to be improperly suited for in situ measurements however. The dependence of the resonant frequency upon temperature is shown in Figures 7-16. The variability is not constant among the resonant frequency modes as indi- cated by the values of K^ and K2 (Table VIII). However, the Nl-Span-C alloy rod appears to be the least variable and may be suited for in situ research if a proper set of temperature correction tables is developed. Due to problems described previously in Section IV, the results are divided into two groups. Group 1 contains the measurements made with Probe 1 on fresh sediment cores but limited by the problems encountered in probe degradation. Group 2 contains measurements made with Probe 2 in which the cores were noted for dessication and compaction while the measuring viscoelastometer remained in good working condition. The results of the experimental measurements and analyses are pre- sented in Table VI. The magnitude of the real component of rigidity ob- served in these surficial continental terrace clayey-silt sediments range from 1.6 X 10 to 2.1 X 107 dynes/cm . Values of the imaginary components of rigidity range between 2.0 X 10^ to 4.1 X 10 dynes/cm . Corresponding shear wave speeds calculated are between 12 and 57 m/sec. Bieda [2], who obtained cores from the same continental terrace area as those analyzed in this study, computed real components of rigidity from 6.5 X 105 to 7 / 2 2.6 X 10 dynes/cm . Observed imaginary components of rigidity ranged from 2.9 X 105 to S.2 X 10^ dynes/cm^. Corresponding shear wave speeds 24 are from 7 to 40 m/sec giving good correlation to sediments from a sim- ilar area utilizing similar viscoelastometer techniques. The magnitudes of the rigidity are less than those reported by Hamilton et al [10] , who derived a value of 1.6 X 10^ dynes/cm^ for sediments off San Diego util- izing Stoneley wave techniques. This difference in magnitude could be related to three factors [2] : (a) Monterey Bay sediments have been re- cently laid down at high deposition rates, allowing little time for ce- mentation and/or compaction; (b) the rigidity could have been reduced due to sampling techniques; (c) the Stoneley wave technique may be affec- ted by at least the first meter of sediment, yielding an averaging of the results in the upper meter of depth. Other researchers, however, have also obtained similar results confirming the existence of low rig- idities. Anderson and Latham [1], in studying the dispersion of Rayleigh waves due to oceanic seismic activity, had to assume an extremely low sediment shear velocity range of 30 to 120 m/sec for a sediment layer 150 meters thick in order to obtain good correlation between theoretical models and experimental data. Oavies [6], studying the dispersion of waves from explosions, obtained shear wave velocities from 50 to 190 m/sec for the upper 16 meters of sediment in the Indian Ocean. Gallagher and Nacci [7] , using a torsional wave vibration method obtained values of 2.3 X 10? dynes/cm^, results which are similar to those obtained in this report. Shear velocities and the concomitant rigidities cited by other researchers are considerably higher than those discussed here [9] . The probable reason for this involves sediments of firmer structure with stronger interparticle bonds. In addition to rigidity, compressional sound speed, shear wave speed, ratio of sound speed in sediment to that in sea water at 20°C, and the 25 product of density and sound speed squared, pc , were calculated (Tables VI and VII). The values of density, water content, vane shear strength, porosity, mean grain size, and the sand-silt-clay ratios were interpo- lated from the values obtained by Cepek [4] (Tables III and V). Because of the physical breakdown of Probe 1, no certainty in the values of G^ and G2 is known, and the values are only relative when con- sidered against each other. Comparisons of the values of Gj and G2 ob- tained from Probe 2 are also restricted because of the limited number of samples. Although these values are considered to be more reliable be- cause no physical problems were encountered with Probe 2, Gj and G2 are presumed unrepresentative of the cores because of the sediment dessica- tion and compaction noted earlier. When comparing the vane shear strengths obtained from core 6A by Cepek [4] to the values of the shear modulus ob- tained from Probe 1, however, the trends are in keeping with each other. It is suspected, therefore, that while the values of Gj and G2 are not representative of the fresh cores, the trends do depict the relative na- ture of the shear moduli of the sediments. When comparing the real component of rigidity, Gj , to density (Figure 17) , both Group 1 and 2 indicate an increase of G^ with increasing den- sity. There is no definite indication of G^ of Group 1 with porosity (Figure 18) but there is a slight increase of G-^ of Group 1 with decreas- ing porosity. The same trends were noted by Bieda [2] in similar sedi- ments and has also been observed by Hamilton [9] . Higher porosities lead to lower densities and lower dynamic rigidities due to fewer interparticle contacts in sands and clayey silts. An increase of rigidity with increas- ing density may also be caused by an increase in pressure and compaction which increase both properties. When comparing G^ to sound speed (Figure 26 19) no determination of trends is attempted because of the small range of variability of the sound speed among the different core sections. As was also noted by Bieda [2] there is no apparent relationship be- tween G± of either probe and the percentages of sand (Figure 20) and silt (Figure 21) . There is an observable increase in Gi of Group 1 with an increasing percentage of clay (Figure 22) . This is in contrast to the trends noted by Bieda [2] in similar sediments and Hamilton [9] in abys- sal plain and hill sediments. Rigidity in fine-grained clayey sediments is dependent upon the cohesive interparticle contacts and bonds. Sedi- ment analysis of Bieda' s cores showed a high percentage of silt and a medium to low percentage of clay, the opposite of the analysis of the cores studied in this report (Table III). This is one possible reason for the reverse in observed trends. As noted previously, vane shear tests (Figure 23) correlate agreeably with rigidity, with G^ of Group 2 showing much stronger trends than Gj of Group 1. In cores such as 6A where the alternate 14-inch sections were analyzed exclusively with Group 1, similar relationships between vane shear and G\ are evident. Many workers in the field of soil mechan- ics still question the value of vane shear strength measurements but the observed trends of this report add support that there may be more factors common to rigidity and vane shear strength than sediment structure. Comparison between G^ and Pc^ shows an increase with G^ of Group 2 with increasing Pc^. No attempt is made to compare G^ of Group 1 with Pc . This trend is in agreement with Hamilton [9] who considers Pc^ the best empirical index to rigidity in continental terrace and abyssal plain environments. When comparing the trends observed by Bieda [2] and those of this report for similar continental terrace sediments, it appears that like 27 relationships are found except in the case of clay percentage. The trends observed for rigidity, density, porosity, and pc^ lend support to Bieda's assumption that these properties are interrelated and affect the rigidity collectively. The increased clay content of the samples studied is assumed to be a factor in the contrast of observed trends although the influences of cementation and the rate of deposition remain unstudied. The physical degeneration of Probe 1 is also assumed to affect the abso- lute values of Gi. The imaginary component of rigidity, G2, representative of the damp- ing of wave energy, exhibits few visible trends when plotted against the various mass-physical properties. G2 of Group 1 and Group 2 shows no general relationships with density (Figure 25) , porosity (Figure 26) , sound speed (Figure 27) , percent clay (Figure 30) , vane shear strength (Figure 31), or Pc^ (Figure 32). Since these properties represented traits of elastic response this is expected. This same relationship was observed by Bieda [2] . Graphical plots of the values of G2 as a function of percentage sand (Figure 28) show an increase of G2 with increasing percent sand. One possible reason is that the size of the sand particles may correspond to the wavelength of the shear wave and some dispersion may have occurred. Another possibility is that in order for sand particiles to cause an in- crease in the rigidity they must be in close contact. In the case of low sand percentages and high porosities, as is indicated in this report, there is probably little particle to particle contact as the particles are surrounded by the viscoelastic clay, thus increasing the damping of the wave energy [9] . In comparing G2 to percent silt (Figure 29) a sim- ilar increase of G2 with increased silt percentages is observed, a trend 28 also noted by Bieda [2]. Looking specifically at G2 of Group 1 as a function of clay content (Figure 30) a reversed trend of G2 as compared to G^ is observed. Thus, as the elastic rigidity increased with increas- ing silt there is an observable increase in G2 with decreasing percentage silt. If the G2 data are compared with vane shear strength data, the values of G2 gathered with both Group 1 and Group 2 increase varyingly with in- creasing shear strength. Since cohesion is the resistance to shear stresses and is considered to be an inherent property of fine-grained, clayey sediments [9] , this reinforces the indication noted by Bieda that the imaginary component of rigidity is determined by properties related to cohesion and structure. When G2 is graphed against Pc^ no trend is noted for G2 of Group 2. A general increase of G2 of Group 1 with an increase of Pc^ is observed. In a majority of core sections the imaginary component of rigidity is greater than the real component of rigidity. The deviation of the parameters used in computing G2 was considerably larger than those in- volved with Gi causing larger variations in the values of G2 (Table IX). However, the larger G2 values are believed to be caused by the sand and silt concentrations. Those core sections with a large G2 were generally the same sections with an increased sand or silt content. This reinforces the indication that sand and silt directly influence the damping of the wave energy. 29 VI. CONCLUSIONS AND RECOMMENDATIONS Although there is some doubt to the accuracy of the absolute values of the dynamic rigidity, it is concluded that the torsional wave visco- elastometer is probably giving valid results for the rigidities obtained. The assumptions inherent in the analysis of the data collected using this viscoelastometer also seem to be valid. Results correlate well with work done by previous researchers at the Naval Postgraduate School and with magnitudes reported for similar types of sediments by workers at other in- stitutions using other measuring methods. The relationships between the rigidity and other sediment properties appear to be consistent with those reported by other workers. Because this method also yields a value of the imaginary component of rigidity, which is related to the sound absorption process, this method should be developed further for use in sediment analysis. The current model viscoelastometer is unsuitable for in situ work because of the stable laboratory conditions required for sediment analysis. It is recom- mended that further research be done to correct the physical deficiencies which lead to probe failure. After final development of an in situ model, the present probe should be utilized for laboratory correlation analysis of cores brought back from the in situ sample area. Comparison of values of dynamics shear modulus determined by various methods are needed for validation of the viscoelastometer technique. The relationships between rigidity and the other mass-physical properties, particularly sediment components, need further investigation. 30 Coring Station Latitude N Longitude W Water Depth fm(m) Bottom Water Temperature, °C IB 36-45 123-16 1830(3347) 1.592 2 36-48 123-06 1745(3191) 1.803 3 36-48.5 123-20 1802(3295) 1.552 4 36-48 123-25 1900(3475) 1.548 5 36-44 123-28 1955(3575) - 6A 36-40.5 123-17 . 1883(3444) 1.610 7 36-43.5 123-10 1917(3505) 1.630 8 36-44 123-01 1690(3091) 1.559 Table I. Station Location, Water Depth, and Bottom Water Temperature. 31 Core Section 2 Depth in the Core inches 14 Mean Grain Size 0 Sand % Silt % Clay % IB 9.71 1.35 19.64 79.01 4 34 9.66 1.09 20.92 77.99 6 54 7.71 18.38 34.92 46.70 2 2 16 7.91 17.91 28.64 53.45 4 36 9.72 .78 21.51 77.71 6 53 9.71 .40 20.19 79.41 3 1 3 9.11 8.51 22.03 69.82 3 24 7.73 31.40 19.36 49.24 5 44 9.06 3.73 27.79 68.48 4 1 3 9.75 1.01 19.43 79.56 3 23 9.78 .43 19.50 80.07 5 44 9.12 6.22 24.97 68.81 5 1 3 9.53 1.35 22.79 75.86 3 23 9.86 .48 17.19 82.55 6 57 8.99 2.19 31.85 65.96 6A 1 3 9.73 2 8 9.80 .39 18.92 80.69 4 28 9.46 .47 24.86 74.67 6 48 8.64 3.25 37.76 58.99 8 69 8.52 3.25 43.33 53.42 8 1 3 9.70 1.14 ■ 18.75 80.11 3 23 9.59 .53 21.11 78.36 5 43 9.48 .50 24.11 75.39 Table II. Textural Analysis of 6-inch Core Sections 32 Core Section 2 Depth in the core inches 7 Mean Grain Size 0 Sand % Silt % Clay % IB NA NA NA NA 3 20 9.68 1.22 20.28 78.50 5 40 8.68 9.73 27.92 62.35 2 1 8 NA NA NA x NA 3 20 8.81 9.34 25.07 65.59 5 40 9.71 .59 20.85 78.56 3 2 6 8.42 19.95 20.69 59.36 4 26 8.39 17.56 22.57 59.87 4 2 6 9.76 .71 19.46 79.83 4 26 9.45 3.32 22.23 74.45 5 2 6 9.79 .91 19.99 79.10 4 26 9.57 1.05 22.08 76.87 5 40 9.28 1.62 26.96 71.42 7 60 NA NA NA NA 6A 3 12 9.63 .43 21.79 77.78 5 32 9.05 1.86 31.31 66.83 7 52 8.58 3.25 40.54 56.31 7 2 6 9.67 .64 21.78 77.58 4 26 9.35 1.53 23.16 75.31 6 46 8.86 2.15 31.80 66.05 8 2 6 9.64 .83 19.93 70.24 4 26 9.53 .51 22.61 76.88 6 46 NA NA NA NA Table III. Textural Analysis of 14-inch Core Sections (Interpolated) . 33 Core Section 2 Wet Density g/cm^ 1.29 Water Content % Shear Strength psi Porosity 0, 0 IB 141 0.659 76.8 4 1.34 126 0.839 75.1 6 1.46 83 0.792 66.6 2 2 1.27 169 0.198 80.9 4 1.26 168 0.676 78.6 6 1.31 149 0.481 80.0 3 1 1.32 137 0.396 77.3 3 1.33 128 1.024 75.2 5 1.37 116 0.566 74.3 4 1 1.21 210 0.532 83.3 3 1.31 141 0.668 77.6 5 1.39 103 0.501 71.3 5 1 1.21 207 0.260 82.5 3 1.31 142 0.591 77.7 6 1.38 121 0.467 76.3 6A 1 1.21 207 0.504 82.7 2 1.23 163 0.683 77.0 4 1.34 125 0.611 75.3 6 1.37 114 0.450 74.0 8 1.38 101 0.417 70.3 8 1 1.33 211 0.127 84.6 3 1.28 167 0.495 80.8 5 1.30 141 0.586 76.9 Table IV. Mass-Physical Properties of 6-inch Core Sections 34 Core Section 1 Wet Density g/cm^ 1.27 Water Content 0. 0 Shear Strength psi NA Porosity 0, IB NA NA 3 1.31 133 0.749 75.9 5 1.40 104 0.830 70.8 2 1 1.27 NA NA NA 3 1.27 168 0.437 79.7 5 1.29 158 0.578 79.3 3 2 1.32 132 0.710 76.2 4 1.35 122 0.795 74.8 4 2 1.26 175 0.595 80.4 4 1.35 122 0.584 74.5 5 2 1.26 174 0.425 80.1 4 1.33 135 0.550 77.2 5 1.35 128 0.508 76.7 7 1.38 117 0.430 76.1 6A 3 1.28 149 0.647 76.2 5 1.35 119 0.530 74.7 7 1.38 108 0.434 72.1 8 2 1.31 189 0.311 82.7 4 1.29 154 0.540 78.9 6 1.31 134 NA 75.1 Table V. Mass-Physical Properties of 14-inch Core Sections (Interpolated) . 35 Shear Wave Core Section Speed pc^ _^1 ^2_ m/sec (10° g-mVcm -sec ) (10" dynes/cm^) IB 1 46.8 2.75 11.0 25.5 3 21.1 2.71 5.8 .41 5 51.0 3.02 21.1 29.7 2 1 12.1 2.81 7.7 16.9 3 5 3 2 18.1 4 33.7 4 2 15.0 4 18.1 5 2 49.9 4 5 15.6 7 20.0 6A 3 12.2 5 21.5 7 35.3 8 2 41.3 4 20.2 6 57.2 2.81 2.95 2.3 3.9 2.94 2.5 15.1 2.73 1.8 2.2 2.92 4.4 .45 2.72 14.8 27.6 2.93 3.3 .28 4.5 3.4 2.78 2.2 8.3 2.93 4.1 4.7 3.01 1.6 16.8 2.83 4.0 21.9 2.79 5.1 .20 2.83 11.3 41.3 Table VI. Mass-Physical Properties of 14-inch Core Sections 36 Core Section Compressional Wave Speed m/sec *Csed/ r uwater m/sec 0.968 Probable Fractional Error in ^sed IB 1 1471.2 0.00099 3 1471.1 0.968 5 1471.9 0.968 2 1 1487.3 0.979 3 - - 5 1471.1 0.968 3 2 1490.8 0.981 4 1483.8 0.976 4 2 1471.1 0.968 4 1471.2 0.968 5 2 1471.5 0.968 4 - - 5 1471.1 0.968 7 - - 6A 3 1471.2 0.968 5 1471.2 0.968 7 1478.9 0.973 8 2 1472.2 0.969 4 1471.3 0.968 6 1471.7 - 0.968 'Computed water speed is 1519.8 at 20°C and 35 /oo. Table VII. Acoustic Properties 37 Frequency , in kHz Kl (X 10"6) K2 (X 10-4) PROBE 1 7, ,94 24, ,18 41, .01 57, ,98 PROBE 2 7, ,68 24, ,02 59, ,70 PROBE 4 22.1 9.6 2.9 4.8 1.4 4.8 .9 2.7 2.1 4.2 .2 3.7 .5 3.2 7.10 2.3 3.8 22.22 .3 2.6 55.18 .6 3.0 Table VIII. A Values of Viscoelastic Calibration Coefficients K, and Ko. 38 Parameter Standard Deviation a Percent Variation in Kl Percent Variation in K2 PROBE 1 Fvac 0.06 Hz none 0.62 Foil 1.1 Hz 0.01 12.0 Goil 0.05 y mhos 2.0 none Fsed 4.7 Hz Gsed 0.01 u mhos Kl K2 PROBE 2 Fvac Gvac Foil Goil 0.42 (X10"6) 68.0 (X10"6) Fvac 0.20 Hz none Gvac 0.37 y mhos 0.07 Foil 0.81 Hz 0.00 Goil 0.05 y mhos 0.59 Fsed 0.67 Hz Gsed 0.06 y mhos 1 0.44 (X10-6) K2 130.0 (X10"6) PROBE 3 0.14 Hz none 0.35 y mhos 0.04 1.2 Hz 0.01 0.04 y mhos 0.48 1.0 none 4.2 none 0.96 none 8.1 none Table IX. Standard Deviation and Variation of Dynamic Rigidity Parameters. 39 Parameters % r\1 % n2 % Gx % G2 PROBE 1 / Fvac 0.78 0.12 0.12 0.78 Foil .... .... • • • • .... Goil .... • • • • .... Fsed 63.0 6.6 6.6 63.0 Gsed 0.80 1.7 1.7 0.80 Kl 1.9 4.00 4.0 1.9 K2 6.6 0.98 0.98 6.6 PROBE 2 Fvac 2.1 0.77 0.76 2.1 Gvac 0.06 0.13 0.13 0.06 Foil .... .... .... .... Goil .... • • • • .... .... Fsed 17.0 37.0 38.0 17.0 h 23.0 7.4 7.4 23.0 K2 7.1 2.4 2.5 7.10 0.88 2.1 2.1 0.88 Table IX. Standard Deviation and Variation of Dynamic Rigidity Parameters. (Cont . ) 40 Blocked Components Co X AAAA R Motional Components 7J JO' L (B) Figure 1 . (A) Coordinates for Shear Wave Propagation (B) Equivalent Electrical Circuit of Rod-Transducer System. 41 (1) Support Thread and Ground Connection r-<- j Press Fit Seam "^—Electrical Connections 3.81 cm Electrode Electrical Grid Glue Joint ■Shear Transducer 1.27 cm •Viscoelastometer Length: 20.32 cm Diameter: 1.78 cm (2) + (3) Figure 2. (1) Cut-Away View of Viscoelastometer (2) Polarity of Shear Transducer, Field Direction (3) Shear Stress Directions. 42 UJ u OS 2 UJ u < H o a UJ o UJ «i !— t (X S UJ N) i—i U H Z W X < 2 UJ H —3 H p2 D-, t— i a <•-, ;> o c u < y V Q Qi < ^ d C_) o <: H 1 c J> < f -J E- C 2 -J H r -J 1— 1 UJ u -J c/3 «= o UJ x; > 1 o 1 — 1 a < a: C£ CQ UJ j CM z 2 CTi r— 1 5 o UJ .—1 u z UJ u 13 O a> o G o t/i a> OS J-i o m +-> c s •H g- UJ 4-1 O I aJ •H Q O O rH CQ a •H Uh 43 r \i o . o -o.... V /—•"■* r GOO ©cs &t- CO < ©,L I I ©o Z o 10 ©n T ©in ■•. o I Jz^szz; "f ftH- ^3 u^ — -'^s-r.-i U.-n-TO«v,-J i t 1 -i gssas Figure 4. Coring Station Locations (Isobaths in Fathoms) . 44 2 1 57' z d i. CORE 1-B 7, z 4 >- Z 56' Hi CORE 2 T ^47' K^ CORE 3 7L 348" 0i CORE 4 £ z 75-1/2" 7. z 2 CORE 5 I A z 71-1/2" ^ m)L CORE 6-A D 2 Z Z 7C 72' 1 CORE 7 6-in SECTION 14-in SECTION Figure 5. Sectioning of Cores Z Z 5S" CORE 8 45 CLAY CLAY » • € o\»/ SANDY CLAY / / SA ^/ Sll ND „T SILTY CLAY 1 t \ • • \ -"• \ ° ^ \ <£ V / CLAYEY / CLAY CLAYEY \ / SAND SILT \ SILTY SANDY SAND SILT /sand 0/ / SILT V 100 SAND - % SAND CONTENT 0 SILT Figure 6. Plot of Sand/Silt/Clay Ratios for Each Core Section Analyzed. 46 o 1 1 1 — o - - 1 o - — o — — °o — 1 I 1 CM C CM LO I— 1 u o UJ CC ZD H $ UJ D- s PJ o E- 7925 7930 7935 RESONANT FREQUENCY (Hz) 7940 Figure 7. Resonant Frequency as a Function of Temperature Probe 1 , Mode 1 • 47 o W 24150 24160 24170 24180 RESONANT FREQUENCY (Hz) 24190 Figure 8. Resonant Frequency as a Function of Temperature Probe 1, Mode 2. 48 CD 1 1 1 1 — O — — \ o - - o - - i I i i 1 0 0 40960 40980 41000 41020 RESONANT FREQUENCY (Hz) 41040 — U1 _ c LO o w H PJ UJ Figure 9. Resonant Frequency as a Function of Temperature Probe 1, Mode 3- 49 u UJ U4 57920 57940 57960 57980 58000 58020 58040 RESONANT FREQUENCY (Hz) Figure 10. Resonant Frequency as a Function of Temperature Probe 1, Mode 4. 50 D 1 1 I 1 1 1 T~ — o — — o «- — • o — — 1 1 1 1 1 I O O 1 — 7650 7660 ■ 7670 7680 7690 7700 7710 RESONANT FREQUENCY (Hz) Figure 11. Resonant Frequency as a Function of Temperature Probe 2, Mode 1. LO o (JJ o u3 L/5 51 0 1 1 o 1 "T ~r i - O — — o — - o o - ] 1 1 1 i 1 o CM tn o PJ E- PJ a, w E- 23990 24010 24030 24050 24070. 24090 RESONANT FREQUENCY (Hz) Figure 12. Resonant Frequency as a Function of Temperature Probe 2, Mode 2. 52 o 1 1 1 1 1 1 — o — I- * o o — — °o — 1 I 1 1 1 1 o s w a, L.0 59640 59690 59740 59790 59S40. RESONANT FREQUENCY (Hz) 59890 Figure 13. Resonant Frequency as a Function of Temperature Probe 2, Mode 5. 53 0 1 1 1 1 1 — o • — o — — o — — o o — 1 1 1 1 1 cm 7094 7096 7098 7100 RESONANT FREQUENCY (Hz) 7102 o CM — LO u W s: o H LO Figure 14. Resonant Frequency as a Function of Temperature Probe 4 , Mode 1 . 54 O 1 1 1 1 1 1 1 — o - — 0 — — o - — °0 — I 1 1 1 I 1 1 22212 22214 22216 22218 22220 22222 22224 RESONANT FREQUENCY (Hz) Figure 15. Resonant Frequency as a Function of Temperature Probe 4, Mode 2. LO u o PJ OS w o- o H — LO 55 D l 1 1 1 ~r ~r — O — — i o — — ■ o — — o o — l 1 I 1 1 i i 55160 55165 55170 55175 55180 55185 55190 RESONANT FREQUENCY (Hz) o CM — O u o OS < cx •^? Figure 16. Resonant Frequency as a Function of Temperature Probe 4, Mode 3. 56 1 1 i 1 - o GROUP 1 o a GROUP 2 CNJ - o o CM — t — \ B o \0 o i— < LO r— 1 <- o H CD O r-l — a d a • L/> - o o o o° ° cP° o o o 1 1 1 I 1 1.2 1.3 1.4 1.5 1.6 DENSITY (g/cc) Figure 17. G^ as a Function of Density, 57 o to I O GROUP 1 D GROUP 2 CM O CM o CM 6 o R C D C3 lo c?o o PB 60 70 80 90 100 POROSITY (%) Figure 18. G1 as a Function of Porosity, 58 o t-Q O GROUP 1 D GROUP 2 O CM B o R C CJ3 D cP O o o 8 1 J. I 1420 1440 1460 1480 SOUND SPEED (ra/sec) 1500 Figure 19. G^ as a Function of Sound Speed 59 o K5 CM O CM T O GROUP 1 □ GROUP 2 6 o ft O T3 C3 D O O h 10 15 PERCENT SAND (%) o I 20 Figure 20. G^ as a Function of Percent Sand 25 60 E o If) CD s. o u 1 1 f 1 1 1 o GROUP 1 o D GROUP 2 1 i o - O CM LO r— 1 Q - O • i— 1 LO - O o D ° o°o° o o o 1 1 I 1 1 1 15 20 25 30 35 40 PERCENT SILT (%) Figure 21. G^ as a Function of Percent Silt 61 o to O GROUP 1 D GROUP 2 LO c s o to R o m Q o OD 60 65 70 75 80 PERCENT CLAY (%) Figure 22. G^ as a Function of Percent Clay. 62 to E a to g. -d u a o o l j T O GROUP 1 D GROUP 2 1 I D .40 .50 .60 .70 SHEAR STRENGTH (lbs/inch2) .80 Figure 23. G-^ as a Function of Vane Shear Strength. 63 - 1 1 1 1 1 O GROUP 1 o Q GROUP 2 CM - D - 1 — \ e o ■■ 13 O i— t V ' i— ( a - .—1 o i—i D o D O • - m o O D O o I 1 1 1 ..... 1 27 28 pc2 (105 29 30 ? 2 7 ■m^/cm -sec ) 31 Figure 24. Gj as a Function of the Product of Density and Sound Speed Squared. 64 o c CM E o m 0) O C rH CM CM T 7=L D I I . O GROUP 1 Q GROUP 2 o _QtQ i. 1 1.2 1.3 1.4 DENSITY (g/cc) 1.5 1.6 Figure 25. G2 as a Function of Density, 65 1 1 1 \ I O 0 O GROUP 1 0 GROUP 2 c to Q o t — \ 6 o o 1 C i— 1 c CM — - o 1—1 o O O 8 — 1 I Q<9 oO I 1 60 70 80 90 100 POROSITY (%) Figure 26. G2 as a Function of Porosity 66 1 1 1 1 T" o — o O GROUP 1 O GROUP 2 c 1 o a - I — v CM 6 o o • a o C r-l C CN1 ~ — O - o o o o ° o - 1 t i m . 1 1420 1440 1460 1480 SOUND SPEED (m/sec) 1500 Figure 27. G2 as a Function of Sound Speed, 67 1 1 1 I 1 o GROUP 1 o <* Q GROUP 2 c Q o e D c i— < c - c r— I O 0 o o o $► 0 | 1 1 ! 1 10 15 20 PERCENT SAND (%) 25 Figure 28. G2 as a Function of Percent Sand, 68 1 1 i { 1 O GROUP 1 o Q GROUP 2 o to a O — / — \ E o I/) R id a o i— i c — — (Nl c — o o o o o CD o a I I O GROUP 1 D GROUP 2 c c CM CM P o o J .40 .50 .60 .70 SHEAR STRENGTH (lbs/ inch2) J O L .80 Figure 31. 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