8 The bender element (BE) test has been widely used for the characterization of soil speci- mens to determine the dynamic or low-strain shear modulus. However, the actual behavior of the BE inside the soil specimen still remains unknown. The current ASTM standard does not consider the interference of P waves in BE measurements, which can lead to signif- icant errors in the evaluation of shear wave velocities. In this paper, the BE motion inside di erent media is numerically studied through a coupled piezoelectric and solid mechanics nite element (FE) model. The numerical results are calibrated and compared with the real motion of the BE monitored using a high-frequency laser vibrometer. The proposed model successfully captured the measured motion of the BE in the air as well as transparent soils. More importantly, the proposed model provided a method for understating the interactions of P waves and S waves in a soil specimen. Simulated signals for an Ottawa sand specimen showed a good agreement with independent results from resonant column tests. The pro- posed piezoelectric-solid mechanics FE model can be used to study the soil-bender element interaction so that sound recommendations can be given to improve the interpretation of BE tests for di erent soils. Keywords: Bender element test, resonant column test, laser vibrometer, wave 9 propagation, solid mechanics, piezoelectric, nite element method 10 Preprint submitted to Canadian Geotechnical Journal December 11, 2020 1. Introduction 11 The dynamic soil properties play an important role in the design of earthquake-resistant 12 structures and foundations. The bender element (BE) test and resonant column (RC) test 13 are the most popular methods used for the evaluation of dynamic soil properties such as 14 the shear wave velocity at low shear strains. The RC test is used to determine the resonant 15 frequency of a soil column, which is related to the shear wave velocity and shear modulus. 16 However, the RC test is time-consuming and costly in comparison to the BE test. The 17 BE utilizes piezo-ceramic materials for the conversion of an electrical signal into mechanical 18 energy. Two bender elements are placed at the two ends of the soil specimen in which 19 one BE is used to introduce a mechanical impulse and the other one is used to receive the 20 propagating pulse (normally in mV). The BE generates not only S-waves in the direction of 21 their plane but also P-waves in the direction normal to their plane. The P-waves re ected 22 from the cell walls can interfere with the generated S-waves [17]. 23 The behavior of the BE has been studied both numerically and experimentally in the 24 literature. Lee and Santamarina [17] showed that the P-wave re ected from the cell wall in 25 the BE test arrives to the BE receiver earlier than the direct S-wave. The interpretation 26 of the BE test results requires the consideration of the geometry of the specimen, such as 27 the radius-to-height ratio [17]. It is found that the resonant frequency of the BE embedded 28 within the soil specimen depends on the sti ness of the BE, soil sti ness, and stress level 29 in the soil [3, 4]. The BE has been used in a triaxial apparatus for the measurement of 30 anisotropy of ne-grained soils [15]; it has shown that the BE is e ective in measuring the 31 inherent anisotropy resulting from the plastic strain history. Youn et al. [27] compared the 32 BE test with the resonant column test as well as the torsional shear test in sands; the results 33 I am corresponding author Email addresses: liuh3429@myumanitoba.ca (Hongwei Liu), gcascante@uwaterloo.ca (Giovanni Cascante), pooneh.maghoul@umanitoba.ca (Pooneh Maghoul), ahmed.shalaby@umanitoba.ca (Ahmed Shalaby) 2 showed that the values of shear wave velocity determined from the BE test correspond well 34 with the resonant column test as well the torsional test in dry conditions [27]. In saturated 35 conditions, however, the BE test tends to overestimate the shear wave velocity in comparison 36 to the two other tests, which is likely due to the S-wave-P-wave interaction [27]. The sti ness 37 of unsaturated soils was tested with the BE and suction-controlled resonant column tests by 38 Hoyos et al. [12]. A good agreement between the BE and resonant column tests was reported 39 in terms of the measurement in sti ness of silty sand under a suction-controlled condition. 40 Gu et al. [11] compared the results of the BE test with the RC test and the cyclic torsional 41 shear test at various con ning pressures, densities, and degrees of saturation. It was found 42 that the travel time, under saturated conditions, obtained from the BE test is considerably 43 smaller than that from the RC test due to the dispersion of the S-wave. However, this 44 result could be attributed to the strong participation of p-waves observed in BE tests in 45 saturated media. Recommendations in terms of the selection of the impulse frequency is 46 given to reduce the subjectivity in arrival time of S wave [3]; a single sinusoidal pulse is 47 recommended and the travel length-to-wavelength ratio should be high enough to reduce 48 the near- eld e ect. 49 A new BE test setup using a laser vibrometer was used by Irfan et al. [14] to monitor the 50 BE motion inside transparent soils. In this experiment, a transparent granular soil specimen 51 was used, such that the real transmitter response can be measured using the laser vibrometer. 52 The transmitter response inside the soil specimen is very di erent from the input electrical 53 signal. The BE response is not well represented by a cantilever beam as typically assumed 54 in the literature [16, 17, 28]. The BE was inserted in the water, sucrose solutions, mineral 55 oils, and air to account for the e ects of viscosity and density of the uid medium. Such 56 tests were used for the calibration of the BE transducer. The important ndings from such 57 experimental tests are: 1) the input square and step function pulses excite higher modes 58 in comparison to the sine function pulses; 2) the uid density dominates the transmitter 59 3 response more than the viscosity of uid; 3) there are strong p-waves generated in typical 60 BE tests. 61 Several methods have been developed in the past to interpret the signal obtained through 62 the BE test. The rst group is to calculate the shear wave velocity based on the rst arrival 63 time, which includes the start-start method, peak-peak method, cross correlation method 64 and cross power method. The cross correlation method is used to measure the degree 65 of correlation of two signals [26] and the cross power method measures the correlation in 66 frequency domain. However, the arrival-time based methods usually results in subjective 67 and inaccurate interpretation of the shear wave velocity. Currently, it is still unclear which 68 method provides the most reliable results [11]. An automatic shear wave velocity estimation 69 method was developed by Finas et al. [9] by applying the Akaike information criteria, which 70 is e ective for a signal-to-noise ratio (SNR) smaller than 4. This inherent complexity in the 71 analysis of BE tests is clari ed in this paper by the use of calibrated numerical simulations. 72 The behaviour of the BE has been numerically studied in the literature. Arulnathan 73 et al. [1] used the nite element (FE) method to illustrate that the interpretation of the BE 74 signals based on the cross correlation between the input and output signals is misleading due 75 to the e ects of wave interference at the boundaries, the phase lag between the mechanical 76 energy and electrical input and multi-dimensional wave travel issues. A two-dimensional 77 (2D) discrete element method was used by O'Donovan et al. [24] to study the response of an 78 idealised granular material in the BE test. The particle velocity data was used to show the 79 propagation of a central S-wave accompanied by P-waves moving along the sides of the soil 80 specimen. A 2D nite element model was used by Ingale et al. [13] to study the e ect of soil 81 types and frequency on BE measurements. It was shown that the FE analysis is consistent 82 with the S-wave velocity obtained through the peak-to-peak method. 83 An analytical model of piezoelectric BE was developed by Zhou et al. [28] based on 84 the rst-order shear deformation theory by assuming a single rotation angle. An analytical 85 4 modelling approach was also developed based on the beam theory under the quasi-static 86 equilibrium condition, which can be used for the optimized design of piezoelectric bending 87 actuators [7]. A close-form 3D piezoelectric model was developed by Rabbani et al. [25] 88 to investigate the free vibration of piezoelectric hollow cylinder using the transfer matrix 89 method and the state space method. 90 Despite the above-mentioned e orts, the actual behaviour of the BE inside the soil still 91 remains unknown. Currently, there is no standard interpretation of the BE measurements 92 due to the complex wave interaction introduced by the BE within the soil specimens. In this 93 paper, a piezoelectric-solid mechanics model is proposed to study the BE motion in di erent 94 media. The model is validated using the BE motion in the air, transparent soil, and Ottawa 95 sand monitored by a laser vibrometer device. The estimation of the soil parameters such as 96 the shear wave velocity and damping ratio of the Ottawa sand using the piezoelectric-solid 97 mechanics model developed in this paper is then compared with independent experimental 98 data obtained via the conventional RC test. Finally, the propagation of P- and S-waves 99 within a soil specimen due to the BE motion is thoroughly studied and the suitability of 100 empirical methods in estimating the S-wave arrival time is discussed. 101 2. Methodology and Experimental Setup 102 In this work, a FE model of a BE-soil sample is calibrated and veri ed using laser vi- 103 brometer measurements on a) BE on air, b) BE in transparent soil, and c) BE in Ottawa 104 sand specimen. Then, the results are independently veri ed using resonant column mea- 105 surements. The detailed procedure is summarized in Figure 1. 106 5 Figure 1: Flowchart of the detailed procedures for the calibration and veri cation of the BE motion as well as wave analysis within a soil specimen. 2.1. Experimental Setup 107 A general schematic of the experimental setup is illustrated in Figure 2. The soil specimen 108 is vibrated at the bottom via a piezoelectric BE. The generated P wave and S wave contribute 109 to the overall displacement at the top surface of the soil specimen. The BE motion is then 110 studied through the laser vibrometer readings as well as a piezoelectric-solid mechanics FE 111 modeling using two main soil sample con gurations: a) transparent soil to evaluate the BE 112 response and b) an Ottawa sand specimen to evaluate the surface response induced by the 113 6 BE. 114 Figure 2: Con guration of the BE test and the laser vibrometer apparatus The BE response under a sinusoidal electrical impulse is monitored in the air through a 115 laser vibrometer apparatus. Based on the response of the BE, a coupled piezoelectrical-solid 116 mechanics nite element model is calibrated so that the numerical prediction ts with the 117 laser vibrometer measurements. The calibrated model in air is then veri ed against the laser 118 vibrometer readings in a transparent soil. In the end, the calibrated model is used to predict 119 independent vibrations induced by the BE motion within an Ottawa sand specimen under a 120 con ning pressure of 100 kPa. The soil parameters determined through trial and error from 121 the FE model are then compared with the values obtained from a conventional RC test. 122 7 2.2. Experimental Tests 123 In the experimental BE tests shown in Figure 2, a function generator (model HP-33120A) 124 generates the desired input voltage signal applied to the BE transmitter through the steel 125 base. The signal is monitored by the oscilloscope (HP-54645A). The BE transforms the 126 electrical energy into the mechanical energy, which then applies an ultrasonic impulse to the 127 soil specimen. The BE transmitter used in this test protrudes 6 mm into the soil specimen 128 and has a 14 mm by 1.0 mm cross section. The laser vibrometer (polytec, 2013) measures 129 the displacement at a single point. 130 The measurements from the Ottawa sand specimen are used for the further veri cation 131 of the proposed numerical model. The soil sample is 7.0 cm in diameter and 14 cm in height. 132 The density of the dry sand is 1,848 kg/m3. The soil specimen is slightly compacted and 133 covered with a latex membrane to hold the sand in place. A 100 kPa vacuum pressure is 134 applied at the bottom of the sand specimen. The vibration introduced by the BE transmitter 135 was captured through the laser vibrometer on the wall of the soil specimen membrane, as 136 shown in Figure 3. The measurements along the specimen are taken for every 1 cm. A 137 re ective adhesive tape is applied to the membrane wall to improve the signal intensity. 138 8 Figure 3: Vibration measurements through the laser vibrometer for the Ottawa sand specimen Furthermore, the resonant column test, as shown in Figure 4, is performed based on the 139 ASTM standard [2] for veri cation of the BE measurements at 100 kPa. The built-in source 140 in the spectrum analyzer (HP-35670A) is used to apply a sinusoidal sweep input voltage. Due 141 to the limited power in the spectrum analyzer, the power ampli er (Bogen GS-250) is used 142 to amplify the input voltage. Such input current introduces the vibration of the magnets, 143 which in turn induce a torsional excitation in the soil sample. The response of the specimen 144 is recorded in terms of acceleration via the accelerometers (PCB353A78 and PCB 353B65) 145 mounted on the driving plate. The current in the coils and the acceleration are ampli ed and 146 ltered (200 Hz low pass) using a lter ampli er (KrohnHite 3384) before being processed 147 by the spectrum analyzer for the transfer function calculations. The spectrum analyzer 148 calculates the transfer function in real time. The resonance frequency and damping ratio of 149 the soil specimen are computed from the transfer function. 150 9 Figure 4: RC test setup for the measurement of shear wave and damping ratio 3. Theoretical Background 151 The motion of the BE in di erent media can be numerically simulated using a piezoelectric- 152 solid mechanics theoretical framework. Also, the soil is considered as a solid, isotropic, and 153 homogeneous medium. This framework can be then implemented in a FE numerical tool for 154 further analyses. In this section, we will brie y review the assumptions and eld equations. 155 The FE modeling is then discussed. 156 3.1. Kinematic Assumptions 157 The linearized form of the Green-Lagrange strain tensor, "ij, for in nitesimal deforma- 158 tions of solid media (BE and soil) and electrical eld Ef iare described, respectively, as 159 10 follows [21]. 160 "ij=1 2(ui;j+uj;i) Ef i= ;i(1) whereui;jrepresents@ui @xj;uirepresents the displacement vector components of the solid 161 medium in each direction and xjrepresents the coordinates; Ef idenotes the electrical eld 162 vector; is the electric potential. 163 3.2. Constitutive Models 164 The constitutive models that describe the stress-strain and electrical displacement- eld 165 relationships are de ned as: 166 ij=Cijkl"klekijEk Di=eikl"kl+ikEk(2) whereijis the stress tensor; ekijandikdenote the piezoelectric tensor and dielectric 167 permittivity tensor, respectively. Direpresents the electrical displacement. Cijklis the 168 fourth-order linear elastic sti ness tensor described in Equation 3 for isotropic materials 169 [19, 20]: 170 Cijkl=ijkl+(ikjl+iljk) (3) whereC11=C33=E0(1) (1+)(12);C12=C13=E0 (1+)(12);C44=C55=C66=E0 2(1+);E0and 171 are Young's modulus and Poisson's ratio, respectively. It is worth mentioning that the 172 stress-strain relationship for soils can be written as ij=Cijkl"kl. Also, the piezoelectric 173 11 tensoreijk(coulomb=m2) is written as [23]: 174 e=8 >>>>< >>>>:0 0 0 0 e150 0 0 0 e150 0 e31e31e330 0 09 >>>>= >>>>;(4) Piezoelectric materials have the ability to produce an electrical voltage with an applied 175 load; vice versa, motions are generated if an electric eld is applied. Such phenomena is 176 described through the piezoelectric tensor. 177 3.3. Conservation Laws 178 Conservation of the linear momentum for a solid medium (BE and soil) is written as: 179 ij;j=ui (5) whereis the bulk density. 180 Gauss's law is used to describe the conversation of charge in the piezoelectric BE: 181 Di;i= 0: (6) 3.4. Field Equations 182 The governing equations for the BE can be written in terms of the displacement vector 183 uias well as the electric eld vector Ef ias: 184 ui;jj+ (+)uj;jiekij;jEf k=ui eikl;i"kl+ik;iEf k= 0:(7) 12 It should be noted the eld equation 7 is obtained through the conservation of momentum 185 and the conservation of charge, respectively. The coupled eld equations are then solved 186 simultaneously due to the coupling tensor eikl, which represents the inherent properties of 187 the piezoelectric materials. 188 Similarly, the eld equation governing the propagation of stress waves into the soil spec- 189 imen due to the piezoelectric BE motion can be written as 190 ui;jj+ (+)uj;ji=ui: (8) 3.5. Finite Element Modeling and Boundary Conditions 191 The 2D FE method is used to solve the eld equations described above via COMSOL 192 Multiphysics [5]. The triangular element type is used for the analysis. The direct solver is 193 used instead of an iterative solver due to its robust nature [18]. There are 5,145 elements with 194 an average quality of 0.8592 (length to width ratio). Based on a extensive mesh sensitivity 195 analysis, it is found that the numerical results are no longer sensitive to the mesh size if a 196 ner mesh is used. With such settings in COMSOL Multiphysics, the relative error can be 197 controlled to an acceptable tolerance. The mesh distribution in the soil specimen and BE is 198 shown in Figure 5. 199 The BE having a height of 6 mm, a length of 14 mm, and a width of 1.0 mm is shaped 200 as a cantilever beam and contains two layers of piezoelectric ceramic plates with a metal 201 plate in the middle. If the poling direction of these two layers of piezoelectric elements is 202 in the same direction, it would be called parallel and if the poling direction is in opposite 203 directions, it would be called series type. The parallel type needs a low voltage to work, but 204 the series type needs, for the same application, twice the voltage magnitude to work [29, 22]. 205 The parallel BE is used as an transmitter and the series type is used as a receiver [29, 22]. 206 The Ottawa sand specimen has a width of 70 mm and a height of 140 mm. 207 13 Figure 5: Mesh distribution for the BE test A sinusoidal impulse voltage with a frequency of 9 kHz, which is close to the resonant 208 frequency of the BE used in our tests, is applied to the BE (plus sign in Figure 5). A xed 209 boundary condition is applied to the top and bottom of the soil specimen. A xed boundary 210 condition is also applied to one end of the BE transmitter (lower end) and receiver (upper 211 end). The remaining boundaries are considered as a free surface (zero stress) to allow the 212 re ection of stress waves. The interface between the BE and surrounding soil is modeled 213 by meeting the continuity conditions. The initial displacement and velocity of the BE and 214 surrounding soils are set to be zero. The components of the coupling tensor in Equation 4 215 has the following values: e31=5:35 C/m2;e15= 15:78 C/m2ande33= 12:29 C/m2[23]. 216 The density of the piezoelectric ceramic plates used in the BE structure is 7,870 kg/m3. The 217 14 mechanical properties of the piezoelectric ceramic plates and the metal plate used in the BE 218 structure, transparent soil and Ottawa sand are obtained through the calibration procedure 219 by trial and error as presented in Section 4.1 and Section 4.2. 220 4. Results and Discussion 221 4.1. BE Motion Calibration and Veri cation 222 The laser pointer is concentrated at a point with a height of 5 mm shown in Figure 2. The 223 comparison of the BE motion in the air between the experimental measurements and numer- 224 ical predictions in time and frequency domains is shown in Figures 6a and 6b, respectively. 225 As can be seen in Figure 6, the numerical results using the calibrated piezoelectric-solid 226 mechanics model are in good agreement with the experimental measurements. 227 The mechanical properties of the BE are calibrated through trial and error to best t 228 the numerical predictions with the laser measurements. The best matching between the 229 numerical results and the laser measurements was achieved when Young's modulus and 230 Poisson's ratio of the piezoelectric ceramic plates are 65 GPa and 0.3, respectively; Young's 231 modulus of the metal plate used in the BE structure is 243 GPa; and, the damping ratio for 232 the piezoelectric ceramic plates and the metal plate used in the BE structure is 2.5%. 233 15 0.00.20.40.60.81.01.21.41.6-4-2024Velocity (mm/s)T ime (ms) Numerical results Experimental measurements(a) time domain 05 1 01 52 02 53 03 54 00.00.10.20.30.40.50.60.7MagnitudeF requency (kHz) Numerical results Experimental measurements (b) frequency domain Figure 6: Comparison between the experimental measurements and numerical results for the BE motion in the air in (a) time domain, (b) frequency domain. The calibrated FE model is further veri ed through the experimental BE tests on the 234 transparent soil (made of silica and oil) where the BE motion was monitored via the laser 235 16 vibrometer. Under the same impulse voltage, the comparison between the numerical FE 236 results and the experimental measurements of the BE motion are shown in Figure 7 in time 237 and frequency domains. 238 0.00.20.40.60.81.01.21.41.6-1.5-1.0-0.50.00.51.0 Numerical results Experimental measurementsVelocity (mm/s)T ime (ms) (a) time domain 05 1 01 52 02 53 03 54 00.000.020.040.060.080.100.12 Numerical results Experimental measurementsMagnitudeF requency (kHz) (b) frequency domain Figure 7: Comparison between the experimental measurements and numerical results for the BE motion in the transparent soil in (a) time domain, (b) frequency domain. 17 The mechanical properties of the transparent soil used in the FE analysis are calibrated 239 by trial and error. It is found that the speeds of the P wave and S wave in the transparent soil 240 studied in this test are approximately 1,200 m/s and 15 m/s, respectively. The equivalent 241 damping ratio of the transparent soil including the visco-elastic e ect is assumed as 0.3. 242 This damping value is high because it is representing not only the damping of the soil, but 243 also the e ect of the added mass of the transparent soil to the response of the BE Irfan 244 et al. [14]. The very low shear wave velocity derived numerically is due to the fact that 245 the con ning pressure is practically zero in this experiment. The high value of the P-wave 246 velocity is generated because of the saturated conditions. The calibrated piezoelectric-solid 247 mechanics model for the BE motion is still able to capture the motion of the BE in the 248 transparent soil. 249 4.2. BE Motion in Ottawa Sand 250 The BE motion can be directly monitored in the air and transparent soil through the 251 laser vibrometer since the laser light can penetrate into these media. It is not, however, the 252 case for the BE test performed on real soils. Therefore, the displacement at the sides of the 253 soil specimen is monitored instead of the BE itself. The setup for this test can be seen in 254 Figure 3. The original laser measurements at the elevations of 2.5 cm (trace 0) to 13.5 cm 255 (trace 12), with an interval of 1 cm, are shown in Figure 8. This signal is contaminated with 256 the higher resonant modes of the BE motion [17]. Since the applied voltage signal is 9 kHz, 257 the components above 15 kHz are removed through the wavelet synchrosqueezed transform 258 [6]. The components below 15 kHz are obtained through the inverse wavelet synchrosqueezed 259 transform. 260 18 0.00 .51 .01 .52 .02 .53 .0024681012Vp = 400 m/sTrace NumberT ime (ms)Vs = 286 m/s (reflection)Figure 8: Original displacement measurement along Ottawa sand specimen using the laser vibrometer Based on the calibrated piezoelectric-solid mechanics model for the BE, the soil pa- 261 rameters are then modi ed through trial and error to match the ltered laser vibrometer 262 measurements. For example, the comparison between typical numerical and experimental 263 displacements is shown in Figure 9. The best tting was achieved by using the shear wave 264 velocity of 240 m/s, compression wave velocity of 380 m/s (equivalent to a Young's modu- 265 lus of 249 MPa and a Poisson's ratio of 0.17), and a damping ratio of 1% for the Ottawa 266 sand. The determined P and S wave velocities can also be veri ed from the original laser 267 measurements. Two constant slopes (400 m/s and 286 m/s slope) are also clearly visualized 268 in the original displacement measurements, as labeled in Figure 8. The determined P wave 269 velocity (380 m/s) and S wave velocity (240 m/s) were relatively close to the P wave veloc- 270 ity (400 m/s) and S wave velocity (286 m/s) visualized in the original laser measurements. 271 19 Furthermore, the dispersion curves were also computed using the numerical and measured 272 displacement data. Figure 10 shows the comparison between the measured and numerical 273 dispersion curves for both symmetric and antisymmetric modes. The numerical predictions 274 showed a reasonable agreement with the laser measurements in terms of the distribution of 275 dispersion curves. Therefore, the P wave and S wave velocity for the Ottawa sand are 380 276 m/s and 240 m/s, which is veri ed through comparison in displacement measurements and 277 dispersion curves. 278 0.00 .20 .40 .60 .81 .01 .21 .4-1.0-0.50.00.51.00.00 .20 .40 .60 .81 .01 .21 .4-1.0-0.50.00.51.0T ime (ms)Height = 7.5cm Numerical results Experimental measurementsHeight = 5.5cmNormalized DisplacementT ime (ms) Numerical results Experimental measurements Figure 9: Comparison between the numerical results and experimental displacement measured at a distance of 5.5 and 7.5 cm from the bottom 20 024681012141618200100200300400500600Phase Velocity (m/s)F requency (kHz) Numerical results Experimental measurementsS ymmetric ModeA ntisymmetric modeFigure 10: Comparison between the numerical and experimental dispersion curves The BE motion at di erent heights (1 mm, 3 mm, and 5 mm) within the Ottawa sand 279 specimen is shown in Figure 11. The rst cycle in the signals is introduced by the applied 280 voltage due to the piezoelectrical e ects. The vibrations after the rst cycle in the signals are 281 due to the free vibration of the BE and wave re ection within the soil specimen. However, 282 the BE motion is mostly dominated by the applied voltage ( rst cycle). Figure 11 also 283 shows the deformation of the BE. The predominant vibrating frequency and period of the 284 BE transmitter is 8.7 kHz and 0.115 ms, respectively in the Ottawa sand specimen. In the 285 rst 1/4 period, the left piezoelectrical plate is moving relatively upward, which drives the 286 BE to move to the left side. Such a phenomenon was already known in Fredy [10] and the 287 proposed piezoelectric-solid mechanics model can describe it physically and qualitatively. 288 BE does not only have the rst mode of vibration under the applied electrical impulse. 289 As shown in Figure 11, the BE exhibited the second mode of vibration after 0.06 ms (1/2 290 period) since the beginning of the impulse load. 291 21 Figure 11: BE transmitter motion numerically predicted at di erent heights 4.3. Wave Analysis within Soil Specimen 292 The wave propagation is further analyzed through the displacement distribution within 293 the Ottawa sand specimen. The transmitter started the generation of mechanical energy 294 aroundt= 0.31 ms. This reference value is shown by Trigger Point in Figure 12. 295 The S wave velocity determined by the BE is relatively higher than the value obtained 296 by the standard RC test Fam et al. [8]. The S wave velocity in the BE test is commonly 297 calculated as distance divided by the S wave arrival time. However, the arrival time of the 298 S wave is normally empirically selected around the rst peak in the output signal. Based on 299 the determined S wave velocity from previous discussion, the exact S wave arrival time can 300 be located around 0.53 ms relative to the beginning of impulse (as labeled by Trigger Point 301 in Figure 12). The measured response in BE testing is given in terms of voltage (Volts) 302 while the numerical response is given by displacement (nm). These two signals are not 303 expected to be identical as the received signal does not have units of displacement, velocity, 304 22 or acceleration. The comparison between the measured and numerically calculated signal 305 at the BE receiver location is shown in Figure 12. A reasonable agreement between the 306 numerical results and experimental data is achieved. The S wave arrival time is normally 307 selected shortly before the peak in the BE receiver signal. However, it is shown that the 308 S wave arrival time is actually a ected by the interaction with the p-wave. Therefore, the 309 proposed numerical model can be used to improve the interpretation of the e ects of p-waves 310 on BE tests results. 311 0.00.20.40.60.81.01.21.4-1.0-0.50.00.51.0S waveP waveNormalized Receiver SignalT ime (ms) Numerical BE response in [nm] BE response in [V]T rigger Point Figure 12: Comparison between experimental data and numerical results at the BE receiver location (this gure shows a relative comparison; the BE signals are not linearly related to vibration measurements such as displacement, velocity, or acceleration. ) The horizontal displacement contours corresponding to travel times equal to one up to 312 ve excitation periods are shown in Figure 13. The positive and negative displacements 313 are shown in blue and red colors, respectively. A full wavelength is de ned by a positive 314 (red) and negative displacement (blue) wavefronts. For the dominant frequency of 8.7 kHz 315 for the BE vibration in Ottawa sand, the S and P waves' wavelengths can be calculated 316 23 ass= 2:8 cm andp= 4:4 cm, respectively. Two wavelengths are identi ed due to the 317 di erent propagation speeds of P and S waves. In the BE test, the generation of the S wave 318 mode (in terms of its amplitude) is much stronger than that of the P wave mode. Only a half 319 wavelength (blue front) was identi ed for the P wave mode due to its weaker generation in 320 the BE test. Furthermore, P wave attenuates with travelling distance and the identi cation 321 of its full wavelength becomes impractical. Therefore, only a half wavelength of the P wave 322 mode is labeled in Figure 13. After the rst period (1T), the re ection of P and S waves can 323 be visualized clearly. The separation between the P wave and S wavefronts increased from 324 the 2ndand 3rdperiods. The P wavefront arrives at the BE receiver tip sometime between 325 the 3rdand 4thperiods. Similarly, the S wavefront arrives at the BE receiver tip sometime 326 between the 4thand 5thperiods. 327 Figure 13: Horizontal displacement contour in Ottawa sand 4.4. Comparison with RC Test 328 The RC test was performed to validate the shear wave velocity and damping ratio of 329 the Ottawa sand under various con ning pressures, as shown in Figure 14 and Figure 15, 330 respectively. As the con ning pressure increases, the soil specimen gradually gains more 331 strength, which results in an increase in the shear wave velocity. On the other hand, the 332 increasing con ning pressure constrains the movement of soil particles, which is shown as 333 24 a reduction in damping ratio.The relation between shear wave velocity ( vs) and con ning 334 pressure (0) for RC test results can be expressed as : vs= 93:90:22 0. Similarly, the relation 335 for damping ratio ( ) and and con ning pressure ( 0) for RC test results is expressed as: 336 = 407:31:26 0. Under a con ning pressure of 100 kPa, the shear wave velocity and 337 damping ratio are found to be 263 m/s and 1.06%, which are consistent with the values 338 used in the numerical model through trial and error. 339 01 002 003 004 005 006 00100150200250300350400450500Shear Wave Velocity (m/s)C onfining Pressure (kPa)vs = 93.9 σ 0.220 Figure 14: Shear wave velocity measured by the resonant column test on Ottawa sand under various con ning pressures 25 01 002 003 004 005 006 000.00.51.01.52.02.53.03.50 .3560 .3430 .3330 .350.4331.063.02ξ = 407.3 σ −1.260 Damping Ratio (%)C onfining Pressure (kPa)Figure 15: Damping ratio measured by the resonant column test on Ottawa sand under various con ning pressures 4.5. Numerical Study of the E ect of Di erent Poisson's Ratios (Loose Sand, Soft Clay) 340 The wave propagation is also investigated in two di erent soil specimens using the 341 piezoelectric-solid mechanics FE model. In Case 1, the P and S wave velocities are 120 342 m/s and 69.3 m/s (equivalent to a Young's modulus of 22 MPa and a Poisson's ratio of 343 0.25) to simulate a loose sand specimen. In Case 2, a soft clay soil sample with a P wave 344 velocity of 120 m/s and an S wave velocity of 35.2 m/s (equivalent to a Young's modulus of 345 6.7 MPa and a Poisson's ratio of 0.45) is studied. The horizontal displacements at the BE 346 receiver location are shown in Figure 16 for Case 1 and Case 2. The exact P wave travel 347 time is 1.07 ms relative to the beginning of the impulse as labeled by Trigger Point. In Case 348 1, the exact S wave travel time is 1.85 ms relative to the beginning of the impulse. Similarly, 349 in Case 2, the S wave travel time is 3.63 ms. 350 26 0.00.51.01.52.02.53.03.54.04.55.05.56.06.5-3-2-1012340 .00.51.01.52.02.53.03.54.04.55.05.56.06.5-3-2-101234T rigger PointTrigger PointS waveP waveDisplacement (nm)Case 1C ase 2S waveP waveDisplacement (nm)T ime (ms)Figure 16: Horizontal displacement at the receiver location for Case 1 (Poisson's ratio = 0.25, loose sand) and Case 2 (Poisson's ratio =0.45, soft clay) The signal obtained in Case 2 is more contaminated by the P wave because of its slower S 351 wave velocity. The components located between the rst P wave arrival time (1.37 ms) and 352 rst S wave arrival time (3.94 ms) are due to the P wave re ection from the sides of the soil 353 specimen, as illustrated in Figure 18. In this case, the selection of the S wave arrival time 354 is rather challenging and unpredictable based on the empirical methods. However, the real 355 arrival time of the S wave is not closely near to the largest amplitude, as shown in Figure 356 16. 357 27 0.00 .51 .01 .52 .02 .53 .0-30-20-1001020300 .00 .51 .01 .52 .02 .53 .0-30-20-100102030Displacement (nm)T ime (ms)Case 1Displacement (nm)T ime (ms)Case 2Δ t = 0.133 ms Δt = 0.128 ms Figure 17: Horizontal displacement of BE transmitter numerically predicted at a height of 5mm for Case 1 (Poisson's ratio = 0.25, loose sand) and Case 2 (Poisson's ratio =0.45, soft clay) The horizontal displacement of BE transmitter is shown in Figure 17 for both Case 1 358 and Case 2. From Figure 17, it is found that BE transmitter vibrates with the dominant 359 frequency of 7.8 kHz for the BE vibration in Case 1. Corresponding, the S and P waves' 360 wavelengths can be calculated as s= 0:89 cm andp= 1:5 cm, respectively. In Case 2, the 361 dominant frequency of the BE motion is 7.5 kHz and the S and P waves' wavelengths can 362 be calculated as s= 0:47 cm and p= 1:6 cm, respectively. The displacement contours at 363 di erent times (normalized in terms of period) are shown in Figure 19 and Figure 18 for Case 364 1 and Case 2, respectively. It is con rmed from Figure 19 and Figure 18 that the re ected P 365 waves from the sides of the specimen can arrive at the BE receiver location faster than the 366 shear wavefront. Therefore, the wave interactions of P and S waves with soil boundaries can 367 largely increase the complexity of the selection of the S wave arrival time. The traditional 368 empirical methods cannot accommodate the complex nature of wave interaction and may 369 result in misleading predictions of the S wave velocity. 370 28 Figure 18: Horizontal displacement contour in low-sti ness clay for Case 2 (Poisson's ratio =0.45, soft clay) Figure 19: Horizontal displacement contour in low-sti ness sand for Case 1 (Poisson's ratio = 0.25, loose sand) 5. Conclusion 371 In this paper, the response of di erent media to a BE motion is thoroughly studied via a 372 piezoelectric-solid mechanics FE model as well as experimental tests. The numerical results 373 29 are compared with the motion of the BE in the air, transparent soil, as well as the Ottawa 374 sand captured by a laser vibrometer. It is concluded that: 375 •The proposed piezoelectric-mechanical model captures the motion of the BE with 376 sucient accuracy in the air, transparent soil, as well as the Ottawa sand. The best 377 agreement was achieved for the BE motion in the air. 378 •The numerical response obtained by the proposed model is consistent with the laser 379 vibrometer measurement at the sides of the Ottawa sand specimen. Furthermore, the 380 numerical predictions show a reasonable agreement with the laser measurements in 381 terms of the distribution of dispersion curves for both symmetric and antisymmetric 382 modes. 383 •A reasonable agreement between the numerical BE response and experimental BE 384 measurement is achieved at the receiver location. The shear wave velocity and damping 385 ratio obtained through the proposed model are consistent with the ones obtained by 386 the resonant column test. 387 •The proposed numerical method shows that there is a signi cant p-wave/s-wave in- 388 teraction that demonstrates why the empirical methods for the selection of s-waves in 389 BE testing could be incorrect depending on the di erent parameters that a ect the 390 participation of p-waves. 391 •The proposed piezoelectric-mechanical model can be used to study the complex wave 392 interactions, which signi cantly improves the interpretation of the e ects of p-waves 393 on BE test results. The proposed model clearly show that the interpretation of BE 394 measurements in clays could be more challenging because of the strong participation 395 of p-waves on the response of BE. 396 30