Study on Shear Wave Attenuation Laws in Granular Sediments Based on Bender Element Test Simulations

Abstract

The propagation characteristics of shear waves in granular sediments are usually used to assess the dynamic response and liquefaction potential of marine engineering foundations. However, the mesoscopic processes by which the excitation frequency influences the shear wave propagation and attenuation remain unclear. In this study, based on a triaxial bender element (BE) test model, the shear wave behavior in uniform spherical particles was simulated by the discrete element method (DEM). It revealed that the BE excites shear waves in a point source manner and that the propagation processes within a triaxial unit cell assembly follow exponential attenuation patterns. Near the vibration source (10–100 kHz), the attenuation law of spherical wave propagation is dominated by friction slip and geometric diffusion in particles. At 0.7–3.5 wavelengths, the shear waves progressively transition to plane waves, and the attenuation law is governed by boundary absorption and viscous damping. At 2.9–10 wavelengths, near-field effects diminish, and planar wave propagation stabilizes. Higher excitation frequencies enhance friction slip, boundary absorption, and viscous damping, leading to frequency-dependent attenuation. The granular system exhibits segmented filtering, with cutoff frequencies dependent on the receiver location but independent of the excitation frequency.

1. Introduction

With the construction and development of marine engineering projects such as artificial islands and offshore wind farms, their engineering safety and durability have attracted increasing attention. Sandy sediments (e.g., calcareous sand, coral sand) serve as the primary filling foundation for island reefs, and their reliability directly impacts the safety and stability of engineering structures [1,2,3]. The shear-wave properties of soils are commonly used to evaluate the dynamic response and liquefaction potential of foundations, providing a non-destructive measurement and prediction tool for the seismic design of engineering structures. Therefore, in-depth investigation into the propagation characteristics of shear waves in sandy sediments holds significant importance for artificial island and reef constructions.

Currently, the methods for assessing the shear wave properties of soil in the laboratory mainly include the torsional resonance column (RC) method, torsional cyclic load method, quartz crystal transducer pulse method, and bending element (BE) measurement method [4,5,6,7]. Among these methods, the BE method stands out because of its distinct testing principle and straightforward operation. A BE probe can be easily integrated into other testing apparatuses for collaborative work, offering significant advantages.

It is well established that sand is a typical granular medium, and granular medium systems exhibit properties such as dispersion, dissipation, frequency dispersion, and sensitivity [8]. The propagation process of waves in a granular medium is affected by complex contacts and interactions among particles, accompanied by strong attenuation phenomena and differences in macroscopic responses at different frequency ranges [4,8,9,10]. The physical experiment (e.g., the frequently used BE test) cannot directly observe the specific response process inside sandy sediments, and it remains challenging to uncover the underlying mesoscopic mechanical mechanisms. The discrete element (DEM) numerical simulation method effectively addresses this limitation. DEM analysis for the BE test can replicate the mesoscopic reaction of shear wave propagation and elucidate the internal mechanisms of the macroscopic response in granular medium systems.

Donovan [11,12] used the DEM to simulate the propagation of shear waves in a regular and randomly arranged system of uniformly sized particles based on BE tests. This study systematically summarized four effective methods for visualizing the mesoscopic response and revealed the propagation characteristics of shear waves within the corresponding period of the received signal through analysis of the mesoscopic response. Additionally, a method for determining the initial arrival time by decomposing the received signal was proposed. Ning et al. [13,14] investigated the effects of the particle size, confining pressure, and excitation frequency on the shear wave velocity within a particle system. By observing the distribution of the particle velocity vector field, the microscopic response state of the simulated specimen was assessed, and the changes in the fabric of the specimen before and after shear wave excitation were compared. Xu et al. [15] employed the cross-correlation method to determine the propagation time of shear waves in DEM simulations, assessed the propagation quality, and discussed the effects of variables such as the excitation source and receiving source dimensions, damping ratio, excitation frequency, and propagation distance on the wave velocity. Gu et al. [16] simulated the propagation process of shear waves excited by BEs in square samples with two receiving points inside the samples. They analyzed the effects of propagation distance, particle size, porosity, input frequency, and damping configurations on the received signals at different positions. The received signal frequency was noted to increase with the excitation frequency until it reached a threshold. They suggested that when simulating wave propagation processes, viscous damping and local damping should be less than 5% and 20%, respectively.

In summary, previous studies have primarily focused on the macro/mesoscale investigations of shear wave velocity, while research on shear wave attenuation and its frequency dependence remains relatively limited. This study establishes a triaxial numerical model of laboratory BE tests using the DEM software PFC3D 5.0 (Particle Flow Code 3D) to investigate shear wave propagation and attenuation in granular media. The macro- and meso-level laws of shear wave propagation attenuation are compared, and the attenuation mechanisms at various propagation stages are explored, thus revealing the filtering effects in sandy sediments under varying excitation frequencies.

2. The BE Test and DEM Simulation

2.1. Integrated BE and RC Test

Bender elements, typically composed of two bonded piezoelectric ceramic plates, are capable of free longitudinal expansion and contraction under an applied voltage. During assembly, the two plates are connected in such a way that when one plate elongates, the other contracts. This differential motion induces bending deformation in the bender element, generating lateral mechanical oscillations—thereby producing shear waves.

The time-domain method for assessing wave velocity in a BE test often exhibits significant subjectivity [17]. Therefore, the experiments in this study employ a combined testing apparatus consisting of a Stokoe-type RC and BE produced by GDS (Geotechnical Digital Systems) in the UK, which effectively overcomes this limitation [18]. The main parts of the combined test equipment also include a confining-pressure controller, back-pressure controller, data collection system, signal amplifier, drive system, and so on. To eliminate the effects of particle shape and size variations in sandy sediments and to focus on the primary mechanisms of shear wave propagation, equal-sized spherical quartz glass beads were selected as the test material. With a 0.3 mm thick latex film as the boundary of a specimen, the glass beads were stacked into a solid cylindrical specimen with a diameter of 50 mm and a height of 100 mm, which was formed in five layers. Figure 1 shows that the stimulating end and the receiving end of the BE were integrated into the centers of the RC specimen base and cap, respectively. Quartz glass beads were formed into cylindrical samples with a diameter of 50 mm and a height of 100 mm. The BE test begins with consolidating the specimen for one hour under a confining pressure of 100 kPa, then the top bending element was excited at different frequencies and the received signal was read through the bottom bending element, and then the RC test was performed. The received signals from the BE test were calibrated on the basis of the results of the RC test to select a suitable excitation frequency and method for analyzing the shear wave propagation time [7]. The fundamental physical property characteristics are presented in

Table 1. Basic physical properties of the quartz glass bead samples.

Specific Gravity of Particle (Gs)	Minimum Dry Density (g/cm3)	Maximum Dry Density (g/cm3)	Particle Diameter (d/mm)
2.23	1.212	1.663	2

.

2.2. DEM Model

The DEM model is based on the actual parameters of the sample in the physical test. Considering the error caused by the thickness of the BE embedded in the RC specimen cap and base during the sampling process, a proportional cylindrical numerical specimen with a diameter of 50 mm and a height of 98 mm was generated. The size and number of particle units were consistent with those used in the physical tests (Φ2 mm, 28,911 pieces), with a corresponding pore ratio of 0.399. Rigid upper and lower pressure plates were set as vertical constraints, and radial constraints were provided by stacked cylinders [14]. This method can simulate rubber film action in physical tests and reduce lateral boundary reflections. The particle–particle and particle–wall interactions were modeled using a simplified Hertzian contact model [19], a nonlinear contact formulation that calculates normal forces through Hertz theory and tangential forces via Mindlin–Deresiewicz theory. This implementation explicitly excludes tensile forces and contact bonding, with contact stiffness governed by the shear modulus and Poisson’s ratio [20]. The normal contact stiffness and tangential contact stiffness are calculated from the following equations [19].

$$K^n={\displaystyle \frac{2G\sqrt{2R}}{3\left(1-\upsilon \right)}}\sqrt{U^n}$$

(1)

$$k^s=\left({\displaystyle \frac{2{\left(3G^2\left(1-\upsilon \right)R\right)}^{\frac{1}{3}}}{2-\upsilon }}\right){\left|F_i^n\right|}^{{\displaystyle \frac{1}{3}}}$$

(2)

where G is the shear modulus, υ is Poisson’s ratio, Uⁿ is the particle contact overlap, and $\left|F_i^{\mathrm{n}}\right|$ is the normal contact force.

A servo-control system maintained specified confining pressure conditions throughout the simulations. The servo mechanism is based on the following algorithm:

$$u_w=g\left({\sigma }^{measured}-{\sigma }^{required}\right)=g\Delta \sigma$$

(3)

where $u_w$ is the wall displacement, g is the stability determination factor, ${\sigma }^{measured}$ is the measured contact stress, and ${\sigma }^{required}$ is the target stress.

During shear wave propagation through particulate systems, particle collisions, translational motion, and rotational motion are induced [21]. While the Hertzian contact model effectively simulates collision and translational behaviors, its inherent inability to transmit torque may lead to overestimation of attenuation caused by particle rotation. However, this limitation has a negligible influence on the overall attenuation patterns.

It should be noted that while the Hertzian contact model is suitable for simulating interactions between non-cemented particles, marine sediments frequently exhibit interparticle bonding behavior. The discrete element modeling of such bonded particle contacts requires further investigation. Additionally, the stacked-cylinder method for flexible boundary simulation inherently cannot represent the “end-clamping effects” in triaxial specimens, which significantly influence the stress distribution and specimen configuration during both compression and shearing [22]. Consequently, this approach is unsuitable for studying shear wave propagation under compressive and shear loading conditions. Since compression and shear effects are not considered in this work, the associated limitations are not taken into account.

In the simulation, a point source was designated for the excitation of the shear wave, utilizing a sine wave as the excitation signal. The shear wave excitation was simulated by inducing transverse vibrations in individual particles. The particles near the top and bottom boundaries were identified as the excitation end of the BE (T) and the reception end (R4), respectively. Additionally, three vibration reception points (R1–R3) were evenly spaced along the line connecting these two points to facilitate the observation of shear wave propagation, as illustrated in Figure 2.

2.3. Calibration of Model Parameters

The model uses a simplified Hertz contact model defined by the shear modulus G, Poisson’s ratio v, friction coefficients, and damping coefficients. G and ν are provided by the quartz glass bead manufacturer and are used directly in the contact model, so only the friction coefficients and damping coefficients need to be calibrated.

The initial jump time of the BE in the physical test and DEM simulation serves as the benchmark for calibration errors. By modifying the friction coefficients and damping coefficients in the DEM model, the error between the jump time in the numerical simulations and that in the physical tests was reduced to less than 5%. The methods used for calibrating the parameters were previously reported [20]. The relevant parameters of the DEM numerical model are presented in

Table 2. Related parameters of DEM simulation.

Parameter	Value
Contact model	Hertz model
Particle size d/mm	2
Proportion Gs	2.3
Shear modulus G/GPa	25
Poisson’s ratio ν	0.2
Damping	0.08
Fric	0.31
Confining pressure/kPa	100
Excitation frequency/kHz	20
Excitation amplitude/mm	1.25 × 102

. Figure 3 shows a comparison of received signals between the physical test of the BE and the DEM simulation at a confining pressure of 100 kPa and an excitation frequency of 20 kHz. The error of the initial arrival wave take-off time point is less than 5%, indicating that the parameter set is suitable for the DEM simulation of the BE test.

3. Analysis of Shear Wave Attenuation Laws

3.1. Macroscopic Attenuation Response

The propagation process of the shear wave from the excitation point T to the bottom receiving point R4 in the numerical test is observed when the excitation frequency is 20 kHz. The signals received at R1, R2, R3, and R4 are shown in Figure 4, reflecting the displacement of the particles along the x-direction at each receiving point. Figure 4 shows that each receiving point generates vibrations sequentially. The near-field effect gradually weakens with an increasing propagation distance at R1, R2, and R3 and is barely noticeable at R3. No near-field effect is observed anymore at the R4 receiver point. The severity of near-field effects is governed by the wavelength number, expressed as R_d = d/$\lambda$, where d represents the propagation distance and where $\lambda$ = V/f denotes the wavelength (with V being the wave velocity and f the input signal frequency), yielding an R_d value of 3.3 in this case. The initial arrival time point of the excited shear wave at each receiving signal zero point was taken as the initial arrival time at different receiving points, followed by the first peak appearing as the initial arrival peak.

3.1.1. Analysis of R1~R4 Received Signals

The amplitudes at the four receiving locations vary from 10⁻¹⁰ m to 10⁻⁸ m, which is much smaller than those at the excitation point T, indicating that the shear wave has attenuated most of the energy when it reaches R1. The initial peak waveforms at each receiving point are distinct, with gradually increasing peak amplitude attenuation, but the rate of attenuation decreases progressively. The trend of peak amplitude attenuation follows an exponential pattern. The normalized first-arrival peak amplitude is denoted as $\overline{S}=S/S_1$. S is the first-arrival peak amplitude of particle vibrations near the straight line from R1 to R4 during shear wave propagation, and S₁ is the first-arrival peak amplitude at the R1 receiving point. The dimensionless propagation distance is denoted as $\overline{L}=L/D$. L is the shear wave propagation distance, and D is the diameter of the numerical sample. The attenuation of the first-arrival peak amplitude can be expressed as Equation (4) [23].

$$\overline{S}=A{\mathrm{e}}^{-\beta \overline{L}}$$

(4)

where A = 3.62 and β = 2.68. β is the attenuation factor that reflects the extent of shear wave attenuation; a larger β indicates greater attenuation of shear waves during propagation. Figure 5 shows the fitted curves of the initial wave attenuation at an excitation frequency of 20 kHz.

3.1.2. Analysis of the T~R1 Received Signal

The shear wave dissipates the majority of its energy upon reaching R1. To observe the specific attenuation of the shear wave at R1, three additional receiving points, Ra, Rb, and Rc, were inserted at equal intervals between T and R1, as shown in Figure 6. The Ra receiving point is adjacent to the excitation point T, whereas the Rb and Rc receiving points are evenly located between Ra and R1. The response of the receiving point caused by the propagation of the shear wave from T to R1 is shown in Figure 7.

There are two distinct intervals of attenuation magnitude between the excitation point T and the reception point R1. The first interval occurs during the transmission of the shear wave from T to the first receiving point Ra, where the shear wave loses the majority of its kinetic energy shortly after initial excitation, resulting in an instantaneous reduction in peak displacement from 10⁻² mm to 10⁻³ mm. The second interval lies between points Rb and R1, where the displacement further diminishes from 10⁻³ mm to 10⁻⁵ mm. Within these intervals, the shear wave experiences significant energy dissipation over a short distance. Figure 8 presents the fitting results of peak amplitude attenuation between receivers Ra and R1 based on Equation (4), demonstrating that the attenuation pattern also conforms to an exponential attenuation pattern. The final equilibrium positions of the particles at three receiving points, Ra, Rb, and Rc, all exhibit varying degrees of displacement, with particles closer to the excitation point T showing a greater degree of displacement, indicating slip between particles in the vicinity of the excitation point. The evolution process of the contact slip zone is shown in Figure 9.

Figure 9 illustrates that the contact slip number reaches its maximum after one cycle of vibration at the excitation point, with a primary concentration occurring at this point initially. As the shear wave propagates, the slip zone gradually spreads downward. However, the number of contacts experiencing slip decreases rapidly, and the extent of slip diminishes. At this time, the energy dissipation due to frictional slip primarily occurs in the segment from T to R1.

3.2. Mesoscopic Attenuation Response

The velocity field of the particles in the XZ cross-section of the DEM model during shear wave propagation was observed. The velocity magnitudes of all the particles in the cross-section were recorded every half cycle (25 μs), as shown in Figure 10. The loss in the average amplitude of the velocity field reflects the superposition of various energy dissipation mechanisms during the propagation of shear waves, visually illustrating the attenuation process from excitation to reception at R4. The dissipation processes mainly arise from particle–particle and wall–wall interactions, as well as from geometrical effects during the propagation of spherical waves. These attenuation mechanisms include viscous damping, frictional slip, boundary absorption, and geometric diffusion [12].

Figure 10 shows that the shear wave generated by the BE as a point source initially propagates to the far end as a spherical wave. The arrival time of the wave front at each receiving point aligns more accurately with the jump points of each received signal depicted in Figure 4. During the initial excitation phase, the peak kinetic energy input occurs near the excitation point T, causing frictional slip between particles adjacent to the excitation source, thereby facilitating energy dissipation. As the propagation distance increases, the wavefront surface continues to expand until it encounters the boundary. After reflection, the wavefront further extends, forming a plane wave. Throughout this process, the geometrical diffusive attenuation mechanism of the shear wave progressively diminishes, resulting in an increase in the overall attenuation amplitude.

Figure 10b–d show that the shear wave propagates downward near the excitation end in the form of a spherical wave, with the wavefront continuously expanding. At a certain moment between Figure 10c,d, the wavefront collides with the boundary, whereas the shear wave reaches reception point R1. The kinetic energy of the red particles with the highest velocity at the boundary is absorbed by the boundary reflection, whereas the velocity decreases as the wavefront expands. Combined with the slip transformation near the excitation end shown in Figure 9, it is evident that the attenuation in the segment from T to R1 is caused primarily by frictional slip, boundary absorption, and diffusion attenuation.

Figure 10d–f illustrate that the shear wave persistently interacts with the boundary within the propagation zone from R1 to R2. As the wavefront expands and the reflected wave interacts with the original wave, the spherical wave transforms into a plane wave. When the shear wave reaches the receiving point R2, the absorption and dissipation of energy by the boundary reflection, along with the attenuation caused by viscous damping between particle contacts during velocity transmission, reveal the intrinsic mechanism of macroscopic amplitude attenuation in this zone. Furthermore, Figure 10g–j illustrate that during this period, attenuation is caused primarily by the viscous damping effect between particle contacts, whereas boundary absorption gradually weakens, which is consistent with the attenuation curve in Figure 5.

A comparison of macro- and mesoscale attenuation laws indicates that during the propagation of shear waves from excitation point T to R1, the geometric diffusion attenuation of the spherical wave and friction dissipation from contact slip near the excited particles are prominent. In the subsequent propagation segments, the geometric diffusion attenuation diminishes. Between reception points R1 and R2, attenuation is caused primarily by boundary absorption and viscous damping dissipation. Between receiving points R2 and R4, the attenuation is due mainly to viscous damping.

4. Frequency Dependence of Shear Wave Attenuation

The propagation of shear waves through a medium is fundamentally characterized by energy transfer and dissipation, which are predominantly governed by interparticle interactions in granular media systems. Extensive studies have revealed that granular media systems inherently exhibit dispersion properties, manifesting distinct macroscopic responses under varying excitation frequencies [12,15]. This suggests that changes in frequency significantly affect the mechanisms of energy transfer and dissipation at the mesoscopic level of contact interactions.

4.1. Effect of the Excitation Frequency on the Macroscopic Response

4.1.1. Attenuation Factor β

Numerical experiments were conducted at different excitation frequencies, during which displacement data at reception points R1 through R4 were recorded. The peak displacement attenuation of the first-arrival wave at these reception points was subsequently fitted in accordance with Equation (1), as illustrated in Figure 11. The attenuation of shear waves progressively decreases with an increasing propagation distance across all the excitation frequencies. As the excitation frequency increased from 10 kHz to 100 kHz, the attenuation process between reception points R1 and R4 was observed to follow an exponential attenuation pattern, with the attenuation factor β increasing from 1.34 to 3.14. This indicates that an increase in the excitation frequency leads to a more pronounced attenuation of shear waves over the propagation distance. When the excitation frequency was below 20 kHz, the variation in the attenuation factor β increased significantly with an increasing excitation frequency. Among these, the attenuation process between reception points R1 and R2 was found to be the most sensitive to the increase in excitation frequency. When the excitation frequency exceeded 20 kHz, the effect of its increase on the variation amplitude of the attenuation factor β diminished, indicating a transition toward a stable critical state.

4.1.2. Frequency-Domain Received Signals

Time-domain signals are superposed multiple sinusoidal waves with varying frequencies, amplitudes, and phases. The bandwidths contained within different excitation frequencies were observed to differ, and the energy corresponding to each frequency component was found to vary. The attenuation of different frequency components in the excitation signal was regarded as a significant manifestation of shear wave propagation attenuation. The time-domain signals of vibrations at reception points R1 to R4 within a 1 ms interval were transformed into frequency-domain signals via the fast Fourier transform, which facilitates the observation of the specific attenuation characteristics of each frequency component along the propagation distance. The frequency-domain signals of vibrations at reception point R4 under excitation frequencies of 10 kHz, 20 kHz, 50 kHz, and 100 kHz are presented in Figure 12.

As illustrated in Figure 12, the frequency-domain representation of the sinusoidal peak is characterized by a dominant frequency band peak with a significant amplitude and a series of secondary peaks exhibiting progressively decreasing amplitudes. The amplitudes within the secondary frequency bands are observed to be substantially smaller than those within the dominant frequency band. The frequency corresponding to the dominant peak is identified as the excitation frequency, and the bandwidth of the dominant frequency band is twice the excitation frequency. The frequency band of the received signal is distributed within the frequency range of the excitation signal. At an excitation frequency of 10 kHz, nearly all frequency components were received almost entirely at reception point R4, with the peak amplitude occurring at approximately 17 kHz. As the excitation frequency increased, the frequency band range of the excitation point also expanded. However, the frequency components at the reception point were confined within a cutoff frequency, demonstrating a pronounced filtering effect. This finding indicates that only a portion of the frequency band from the excitation signal could be received at the reception point, and the amplitude contributions from frequency components exceeding the cutoff frequency were negligible compared with those at lower frequencies. Furthermore, higher excitation frequencies resulted in a broader frequency band range within the excitation signal, whereas the cutoff frequency remained unchanged. Consequently, the amplitudes of the receivable frequency components were observed to progressively decrease, and the filtering effect became increasingly significant.

The frequency-domain received signals at reception points R1 to R4 under their respective excitation frequencies are presented in Figure 13. In the figure, under the same excitation frequency, the frequency band range received by reception points R1 to R4 is shown to gradually narrow, and the frequency band range attenuated between reception points is also observed to progressively decrease. The filtering effect is demonstrated only when the frequency band width of the excitation signal exceeds the cutoff frequency, with the cutoff frequencies being identified as 63 kHz, 37 kHz, 30 kHz, and 28 kHz. This indicates that high-frequency components of the excitation signal exceeding 63 kHz are rapidly attenuated within the segment from T to R1, followed by the sequential attenuation of frequency components above 37 kHz and 30 kHz in the segments from R1 to R2 and R2 to R3, respectively. Consequently, the frequency components of the signal that are ultimately received by R4 through the numerical specimen are found to be below 28 kHz.

As the excitation frequency is increased, the amplitudes at all reception points are shown to exhibit a decreasing trend, whereas the cutoff frequency remains unchanged. The dominant frequencies corresponding to the peak amplitudes are located approximately within the range of 11 kHz to 18 kHz. The dominant frequencies of R1 and R2 are observed to exhibit significant differences between excitation frequencies of 20 kHz to 100 kHz and 10 kHz, whereas those of R3 and R4 are found to show minimal variation with frequency. This suggests that the contact stiffness and mass distribution near the reception points are relatively similar and that the high-frequency excitation signal is observed to affect the contact distribution in the segment prior to reception point R2. This phenomenon is attributed to the contact slip of the particles near the excitation source T. Furthermore, for excitation frequencies exceeding 20 kHz, the effect of the excitation frequency on the contact distributions of R1 and R2 is shown to exhibit minimal differences.

In general, granular systems are characterized by segmented filtering behavior, where the cutoff frequency is determined by the location of the reception points and is independent of the excitation frequency. The natural frequency is an intrinsic property of the granular system, which is unrelated to the excitation frequency but is affected by the contact stiffness and mass distribution near the reception points. High-frequency excitation signals affect the contact distribution around reception points closer to the excitation source; however, this effect gradually diminishes with an increasing propagation distance.

4.2. Effect of the Excitation Frequency on the Microscale Attenuation Process

A comparison was conducted on the velocity field variations and contact slip conditions in the XZ cross-section near each reception point when the shear wave first arrived under excitation frequencies of 10 kHz, 20 kHz, 50 kHz, and 100 kHz, as illustrated in Figure 14 and Figure 15. This comparison provides a visual representation of the effects of frequency variations on the primary attenuation mechanisms of shear waves across different propagation segments.

Overall, an increase in the frequency of the excitation signal was observed to increase the kinetic energy input at the excitation source, resulting in a significant increase in the wavefront diffusion speed and a reduction in the arrival time of the first-arrival wave at each reception point. However, the average amplitude of the particle velocity field in the cross-section was found to decrease significantly with an increasing frequency. The response processes observed in each propagation segment indicate that an increase in the excitation frequency has a significant effect on the attenuation of shear waves in the segment from T to R2. On the one hand, the interactions between particles and boundaries are intensified during T to R2 propagation, leading to an increase in viscous damping attenuation and boundary absorption attenuation. On the other hand, the high-frequency excitation signal was observed to expand the particle slip zone around excitation point T, which diverges spherically downward, thereby enhancing the effect of frictional attenuation mechanisms. However, owing to the constraints imposed by boundary absorption and viscous damping, the slip zone is not allowed to expand indefinitely but is confined primarily to the segment between T and R2. This reflects, to some extent, the effect of the excitation frequency on the coupling between the friction dissipation, boundary absorption and viscous damping attenuation mechanisms, i.e., the high-frequency excitation signal will strengthen the coupling of the attenuation mechanism at the early stage of shear wave propagation. At the same time, combined with the relationship of the attenuation factor with frequency in Figure 10, we can infer that there is a threshold for the coupling of the attenuation mechanism that is strengthened by the increase in the excitation frequency, and the macroscopic attenuation response does not change much when the excitation frequency is increased above this threshold.

5. Discussion

5.1. Near-Field Effect

Figure 4 reveals the occurrence of negative polarity during shear wave propagation, which can be attributed to the near-field effects generated in the propagation process. The near-field effect is typically identified through the negative polarity induced in the first arrival of shear waves by compressional waves. The influence of near-field effects can be significantly reduced when the receiver is positioned at a distance of at least two to five wavelengths from the source [24,25]. As demonstrated in the preceding analysis, the near-field effects become virtually negligible at an excitation frequency of 20 kHz when the normalized distance R_d > 3.3. Figure 16 presents the time-domain waveforms recorded at monitoring points where near-field effects diminish under excitation frequencies of 10 kHz, 50 kHz, and 100 kHz.

Figure 14 shows that the shear waves propagate in a spherical wave pattern between the excitation point T and receiver R1 across the investigated frequency range, with R_d values ranging from 0.7 to 3.5. As the propagation distance subsequently increases, the spherical wave progressively transitions into a planar wave. In conjunction with Figure 16, a comprehensive analysis is conducted on the phase pull-down phenomenon induced by near-field effects. At an excitation frequency of 10 kHz, residual near-field effects persist when the shear wave propagates to receiver R4, where R_d = 2.9. At an excitation frequency of 50 kHz, the near-field effects diminish completely as the shear wave propagates to receiver R3, where R_d = 7.1. At an excitation frequency of 100 kHz, the near-field effects are fully attenuated as the shear wave propagates to receiver R3, where R_d = 10. Therefore, within the excitation frequency range of 10 kHz to 100 kHz, the propagation mode of shear waves undergoes a transition from spherical to planar waves when the normalized distance R_d satisfies 0.7 ≤ R_d ≤ 3.5. When R_d exceeds 2.9 and reaches 10, the influence of near-field effects on shear wave propagation can be considered negligible, and the shear waves propagate predominantly in a planar wave mode.

5.2. Scope of Application of the DEM Model and Follow-Up Recommendations

In this manuscript, the propagation of shear waves in granular sediments has been successfully modeled using the DEM method, but there are some shortcomings and suggestions for follow-up research.

The Hertz contact model used in this manuscript is defined by the shear modulus and Poisson’s ratio, which can further analyze the influence of key parameters such as Poisson’s ratio and shear stiffness on the near-field effect.

The stacked-cylinder method has been widely used in various studies [14,26]. It has a significant limitation in that the center of each ring is fixed in space, and therefore the ring’s geometry applied a symmetrical deformation boundary condition to each specimen region surrounded by a ring [27] and did not allow the characterization of the clamping effects at the ends of the triaxial specimen. This limited the model in the study of triaxial compression or shear problems. It can be compared with other flexible boundary simulation methods such as the periodic boundary method [28], the equivalent algorithm [29], the bonded-particle membrane [30], and the more recent coupled PFC-FLAC with the shell approach [31] to explore the shear wave propagation in compression and shear process.

The present study focuses exclusively on shear wave attenuation in dry granular assemblies. In practical engineering scenarios, however, sediments typically contain water and gas phases. While numerous scholars have investigated discrete element method (DEM) models for both saturated and unsaturated soils [32,33], future work should develop more appropriate models specifically addressing shear wave attenuation in fluid-bearing particulate systems.

6. Conclusions

A thorough investigation into the shear wave propagation characteristics of granular sediments was identified as essential to gain a comprehensive understanding of the dynamic response and liquefaction potential of the marine engineering foundation. The excitation and reception processes of shear waves in a BE test utilizing silica glass beads were numerically simulated by PFC3D software, with the propagation process of shear waves in sandy sediments successfully reproduced. The relevant parameters in the DEM model were calibrated by comparing the received signals and the first-arrival wave initiation points between the physical experiments and numerical simulations. The macro- and mesoscale characteristics of shear wave attenuation in different segments of the DEM model were observed, and the effect of the excitation frequency was investigated. The main conclusions are as follows:

• In the region proximal to the transmitter, shear waves propagate as spherical waves, transitioning to planar wavefronts with an increasing distance. Across both propagation regimes, the peak amplitude attenuation of first arrivals follows an exponential decay pattern, where the attenuation coefficient serves as a quantitative measure of shear wave energy dissipation.
• In the vicinity of the excitation source, attenuation mechanisms including frictional slip, boundary absorption, and geometric spreading dominate the wave energy dissipation. Within the intermediate propagation distance of 0.7 to 3.5 wavelengths from the source, boundary absorption and viscous damping emerge as the predominant attenuation mechanisms.
• An increase in excitation frequency increases the degree of shear wave attenuation and enhances the coupling effect of frictional slip, boundary absorption, and viscous damping attenuation mechanisms in the T to R2 region. Furthermore, high-frequency components of the excitation signal exceeding the cutoff frequency are observed to exhibit segmented filtering characteristics, undergoing rapid attenuation over short distances. The cutoff frequency is determined by the location of the reception points and is independent of the excitation frequency.