The effect of varying stiffness in horizontal structural planes on stress wave propagation is analyzed. Stiffness values of 2.8 × 10⁴ Pa, 2.8 × 10⁵ Pa, 2.8 × 10⁶ Pa, 2.8 × 10⁷ Pa and 2.8 × 10⁸ Pa are analyzed.

By varying the stiffness of a single horizontal zero-thickness structural plane, vertical permanent displacement curves for different stiffness values were obtained, as shown in Figure 3. For structural plane stiffness values above 2.8 × 10⁶ Pa, both vertical permanent displacement and its growth rate increase with rising stiffness. When stiffness is below 2.8 × 10⁶ Pa, the changes in permanent displacement and its growth rate are negligible, consistent with observed results. At stiffness levels below 2.8 × 10⁶ Pa, the system behaves like a one-dimensional Hopkinson bar without a structural plane, resulting in minimal change in permanent displacement.

As shown in Figure 4, the y-direction acceleration curves indicate that for structural plane stiffness values exceeding 2.8 × 10⁶ Pa, the acceleration increases with higher stiffness. For stiffness values below 2.8 × 10⁶ Pa, the acceleration curves show minimal change. Higher structural plane stiffness results in greater acceleration values at the endpoint and earlier stress wave propagation, consistent with observed data.

As shown the y-direction stress curves indicate that stress waves are detected earliest at a stiffness of 2.8 × 10⁸ Pa, followed by 2.8 × 10⁷ Pa. For stiffness values below 2.8 × 10⁶ Pa, the stress wave values are minimal, aligning with observed results. Lower stiffness in structural planes induces significant refraction and reflection of stress waves, resulting in a gradual reduction in their propagation speed.

The above discussion reveals that when the structural plane stiffness exceeds 2.8 × 10⁶ Pa, y-direction permanent displacement, acceleration, and stress values increase with higher stiffness. This highlights the role of high-stiffness structural planes in enhancing stress wave propagation. Conversely, when the stiffness falls below 2.8 × 10⁶ Pa, y-direction permanent displacement, acceleration, and stress values remain largely unchanged as stiffness decreases. This aligns with real-world observations and highlights the critical role of stiffness thresholds in stress wave propagation.

By varying the number of structural planes in a one-dimensional rock bar, the propagation behavior of stress waves through different quantities of structural planes is analyzed. As the number of structural planes increases, the spacing between them is kept at 1 m from the centerline.

We will discuss the scenarios of setting up single, two, three and four Zero-Thickness Structural Surface, respectively.

As shown in Figure 5, with a single structural plane, the vertical permanent displacement exhibits the fastest rise and highest peak value. With two structural planes, positioned symmetrically outward from the center of the rock bar, the stress wave encounters the first plane earlier compared to the single-plane scenario. The stress wave undergoes attenuation from refraction and reflection as it passes through the first structural plane and propagates to the second. With three structural planes, attenuation intensifies compared to two; with four planes, propagation slows further, continuing this trend with additional planes. This demonstrates that increasing the number of structural planes amplifies refraction and reflection, resulting in continuous attenuation of the stress wave. As a result, the permanent displacement at the endpoint decreases progressively.

As shown the y-direction acceleration curves indicate that with an even number of structural planes, acceleration is detected earlier on the time axis, particularly at the spectral peak. For an odd number of structural planes, the spectral peak appears later in time. As the number of structural planes increases, the spectral peak shifts further in time, and the average acceleration value decreases progressively. This behavior correlates with the refraction and reflection phenomena, where more structural planes result in greater attenuation of the stress waves.

As shown the y-direction stress curves for stress waves passing through varying numbers of structural planes. The stress curve spectrum is detected earliest with four structural planes, followed by three, two, and finally one. This indicates that fewer structural planes result in a slower arrival of the stress wave at the opposite endpoint. This phenomenon is attributed to the higher stiffness and elastic modulus of structural planes compared to the rock bar, along with energy dissipation during wave transmission. As noted in Section 1.1, weaker structural planes induce significant refraction and reflection, reducing the stress wave’s propagation speed. Increasing the number of structural planes is akin to raising the elastic modulus, aligning with observed results.

By adjusting the spacing of structural planes in a one-dimensional rock bar, we analyze the propagation behavior of stress waves through structural planes with varying spacings. Using the centerline as a reference, structural plane spacings of 1 m, 2 m, and 3 m are configured for analysis.

Using the midpoint of the rock mass as the baseline, structural plane spacings of 1 m, 2 m, and 3 m are chosen above and below to examine the impact of spacing on stress wave propagation. Figure 6 shows that as structural plane spacing increases, the y-direction permanent displacement at the far end of the Hopkinson bar gradually rises, though the increase remains modest. This occurs because increasing the spacing between structural planes weakens refraction and reflection, resulting in higher permanent displacement. At larger structural plane spacings, reflections between planes nearly vanish, aligning with observed conditions.

Figure 7 illustrates that as structural plane spacing increases, the y-direction acceleration at the far end of the Hopkinson bar gradually rises, though the increase remains modest. This is attributed to the reduced refraction and reflection between structural planes as their spacing increases, similar to the explanation for permanent displacement.

As shown the y-direction stress curves for stress waves passing through structural planes with varying spacings. When the structural plane spacing is 1 m, the stress wave reaches the endpoint the earliest. With increasing structural plane spacing, the stress wave arrives at the endpoint progressively later. This occurs because wider structural plane spacing slows stress wave propagation and dissipates energy, resulting in a gradual reduction in the spectral peak value.

By adjusting the thickness of structural planes in a one-dimensional rock bar, we analyze the propagation behavior of stress waves through planes of varying thicknesses. Using the centerline as the reference, structural plane thicknesses of 0 m, 1 m, and 2 m are configured for analysis.

Using the midpoint of the rock mass as the baseline, structural planes with thicknesses of 0 m, 1 m, and 2 m are positioned above and below to examine the effect of thickness on stress wave propagation. Figure 8 shows that as the structural plane thickness increases from zero to 1 m and 2 m, the plane behaves as a weak interlayer, affecting stress wave propagation. As the thickness increases, the displacement and its growth rate over time gradually decrease. This aligns with field conditions, where weak interlayers absorb stress wave energy, enhancing refraction and reflection effects. Thicker weak interlayers amplify these effects, reducing permanent displacement, consistent with the numerical results.

As shown the y-direction acceleration is detected earliest when the structural plane has zero thickness. As the structural plane thickness increases, acceleration is detected later at the opposite end of the rock bar. Increasing structural plane thickness weakens stress wave propagation, affecting acceleration timing and magnitude, consistent with field observations.

As shown the y-direction stress curves, the spectral peak appears earliest for structural planes with zero thickness, while thicker planes delay the peak occurrence. Stress values are detected earliest at the opposite end of the rock bar when the structural plane has zero thickness. As structural plane thickness increases, point B detects stress values at progressively later times. The maximum stress peak decreases as structural plane thickness increases, indicating that thicker planes weaken the stress wave propagation.

The above discussion reveals that greater structural plane thickness enhances its role as a weak interlayer, significantly attenuating stress wave values, consistent with observed conditions.

We will discuss the scenarios where Stiffness values is set to stiffness values of 2.8 × 10⁶ Pa, 2.8 × 10⁷ Pa and 2.8 × 10⁸ Pa, respectively.

Varying the stiffness of the 60° sawtooth-shaped structural plane yielded vertical permanent displacement curves for different stiffness values, as shown in Figure 10 at a stiffness of 2.8 × 10⁶ Pa, the sawtooth-shaped structural plane exhibited negligible permanent displacement. As the stiffness of the sawtooth-shaped structural plane increased, permanent displacement rose accordingly, indicating a positive correlation.

This observation aligns with real-world conditions:low structural plane stiffness behaves similarly to the rock bar material, exerting minimal influence on permanent displacement. Noticeable permanent displacement occurs only when the stiffness of the sawtooth-shaped structural plane equals or exceeds that of the rock bar.

Figure 11 shows that the y-direction acceleration curves of stress waves propagating through sawtooth-shaped structural planes exhibit significantly lower stress and acceleration values when the stiffness is below the normal level (2.8 × 10⁶ Pa), with notable differences observed. This confirms that high-stiffness sawtooth planes enhance stress wave transmission, whereas low stiffness increases wave attenuation and energy dissipation.

We will discuss the scenarios where the angle of the sawtooth structural surface is set to 30°, 60°, 90°, 120°, 150°, and 180°, respectively.

Figure 12 shows that the permanent displacement after stress wave propagation is smallest for a sawtooth-shaped structural plane with a 180° angle (horizontal), indicating that sawtooth structures significantly affect displacement. At a 30° sawtooth angle, the displacement exhibits the highest initial growth rate and peak value. For other angles, displacement growth rates are similar initially but diverge after 0.2 s. The descending order of permanent displacement is 150° > 120° > 90° > 30° > 60° > 180°.

As shown the y-direction acceleration curves exhibit varying spectral peak values depending on the sawtooth-shaped structural plane angle. The smallest peak value occurs at 180°. As the sawtooth angle increases, stress and acceleration values decrease, following the order 180° < 150° < 120° < 90° < 60° < 30°. This trend aligns with actual observations, as larger angles enhance refraction, reflection, and attenuation effects.

The material parameters of the rock bar model and the filling material are shown in Table 2.

As illustrated in Figure 14, a structural plane with a 5 m thickness was subjected to two filling conditions: full filling and half filling, the latter involving a diagonal triangular fill, to analyze the y-direction acceleration curves of stress waves. During the initial 0–0.15 s, the triangular filling method enabled the half-filled case to transmit the stress wave more quickly. As a result, within this interval, the half-filled case exhibited greater displacement and growth rates compared to the fully filled case. After 0.15 s, as the stress wave finishes propagating through the filled region, the fully filled case exhibits a rapid increase in speed, ultimately resulting in a final displacement approximately 300% greater than that of the half-filled case.

As shown the y-direction acceleration curves reveal that at the initial moment, stress and acceleration values are first detected in the half-filled structural plane. Over time, the stress and acceleration values at the opposite end of the rock bar for both fully filled and half-filled structural planes tend to converge, showing upward and downward waveform fluctuations.

Figure 14 and the y-direction acceleration curves demonstrate that fully filled structural planes attenuate stress wave propagation energy more effectively than partially filled or unfilled ones, likely due to the absorption and scattering effects of the filling materials.

By altering the density of the filling materials, we analyzed the propagation behavior of stress waves through materials with varying densities. Table 3 provides the material parameters of the rock bar model and the various filling materials used in the simulations.

As shown in Figure 15, structural planes were filled with materials of varying densities. The permanent displacement at the opposite end of the rock bar increases with higher filling densities. The permanent displacement at 2,000 kg/m³ is notably higher than at 1,800 kg/m³ and 1700 kg/m³, attributed to the reduced influence of the weak interlayer.

The stress and acceleration curves reveal that waveform values appear earliest for a density of 2,000 kg/m³, followed by 1,800 kg/m³, and lastly for the lowest density. This demonstrates that higher density results in faster stress wave propagation.

Figures 15, 16 show that increasing the density of the filling material enhances the propagation speed of stress waves and reduces energy attenuation, leading to higher permanent displacement and acceleration values. This suggests that high-density filling materials improve stress wave propagation efficiency and minimize energy loss.

In static problems, adjusting the damping coefficient ensures convergence to a static state, with convergence speed directly proportional to the damping magnitude. Insufficient damping causes repeated oscillation of mesh nodes (Gagnon et al., 2019). In dynamic analyses, damping represents the energy dissipation in elastoplastic materials, such as rock and soil, during deformation and failure. Damping significantly affects wave propagation and the dynamic response of geotechnical structures. Excessive or insufficient damping can lead to divergent results, failing to represent actual conditions. Thus, selecting an appropriate damping coefficient is essential for ensuring accurate calculation results.

The governing equation for the motion of grid nodes is given by Equation 1: m u ¨ i + c u ˙ i + F i e = F (1)

For the characteristic time, assume T e = 2 π m / K , t = T e t ′ , the governing equation becomes Equation 2: u ¨ i + 2 π c m K u ˙ i + 4 π 2 K F i e = 4 π 2 K F (2)

ζ represents the dimensionless damping ratio. Its expression is given by Equation 3: ζ = 2 π c m K (3)

From Equation 3, we can obtain the expression for c, which is Equation 4, consequently, c = 1 2 π ζ m K (4)

In the formula, m represents mass, c represents damping, K represents element stiffness, F represents force, F _i ^e represents nodal force, T _e represents characteristic time, üі represents acceleration, ủ і represents velocity.

The damping ratio influences the natural frequency and vibration duration of elastoplastic materials such as rock and soil. In dynamic calculations, adjusting the damping ratio modifies vibration duration, allowing the system’s damping ratio to be inversely determined from shaking table experiment results.

We will discuss the scenarios where the damping is set to 0, 0.4, and 0.8, respectively.

As illustrated in Figure 20, by varying damping values to study the stress wave propagation process, displacement curves reveal that higher damping increases energy dissipation, leading to smaller endpoint displacements.

As illustrated in Figure 21, the axial stress and acceleration curves reveal that as the damping ratio increases, stress loss intensifies, resulting in progressively smaller stress and acceleration values. Higher damping ratios accelerate energy dissipation, significantly reducing the stress wave amplitude at the propagation endpoint. This highlights the crucial role of damping in wave energy attenuation, particularly in complex geological conditions where damping effects can critically influence slope stability.

The propagation characteristics of stress waves in rock slopes are governed by various factors, including the characteristics and parameters of structural planes, wave impedance, and damping. These elements collectively contribute to attenuation of stress wave amplitude, signal delay, and reduced wave speed, the physical mechanism is primarily manifested in the following two aspects. Firstly, the complexity of structural planes:in actual rock masses, the characteristics of structural planes are extremely complex, including varying dip angles, fill materials, stiffness, density, undulations, etc. These factors directly affect the reflection, refraction, and transmission of stress waves at the structural planes. For example, at inclined structural planes, stress waves may undergo refraction and reflection, leading to tensile cracking and slippage at the planes; the fill materials within the structural planes may absorb part of the wave energy, reducing the amplitude of the wave after it passes through; fill materials of different shapes can cause both refraction and reflection in addition to reducing the wave amplitude, resulting in more complex mechanical behaviors. Secondly, the interaction between waves and structural planes: when stress waves propagate to a structural plane, part of the wave is reflected back into the original medium, while another part is transmitted into the adjacent medium. This interaction depends on the properties of the wave (such as frequency, wavelength) and the characteristics of the structural plane (such as roughness, continuity). If the structural plane is rough, the wave will undergo multiple scattering during propagation, leading to dispersion and attenuation of wave energy.

Using a one-dimensional Hopkinson bar model, this study systematically analyzed stress wave propagation under various structural plane conditions, monitoring key parameters such as permanent displacement, acceleration, and stress values. The following well-structured conclusions were drawn: