Characteristics of Compressive Stress Wave Propagation Across a Nonlinear Viscoelastic Filled Rock Joint

Abstract

Filled joints significantly influence the dynamic response of rock masses, exhibiting coupled nonlinear compression-hardening and viscous deformation. However, the combined effects of these mechanisms on wave propagation remain unclear. This study develops a theoretical model based on a nonlinear viscoelastic formulation, in which a compression-hardening spring (governed by the Bandis–Barton model, with its initial compressive stiffness and maximum allowable closure) is connected in series with a viscous dashpot. Using the displacement discontinuity method and the method of characteristics, we analyze the transmission of compressive stress waves across a filled joint. The results show that the transmission coefficient increases with incident wave amplitude but decreases with frequency, whereas reflection exhibits the opposite trends. The initial compressive stiffness has a minimal impact on transmission but induces a nonlinear decrease in reflection. Increasing the maximum allowable closure slightly reduces transmission but sharply increases reflection, whereas higher viscous stiffness enhances transmission and slightly suppresses reflection. Energy attenuation grows rapidly with amplitude before stabilizing. The initial compressive stiffness is most influential at low amplitudes, the maximum allowable closure is most significant at moderate amplitudes, and viscous effects remain consistent across all amplitudes. Increases in frequency lead to a nonlinear decrease in attenuation, with the initial compressive stiffness and maximum allowable closure dominating at high frequencies, and viscous effects prevailing at low frequencies. This work systematically reveals the coupled roles of nonlinear compression-hardening and viscosity in wave propagation across filled joints, providing theoretical support for dynamic hazard mitigation and geophysical exploration.

1. Introduction

Rock masses are complex geological media composed of a rock matrix and various discontinuities. These discontinuities, acting as dominant structural interfaces, fundamentally control the mechanical behavior of the rock mass, and their dynamic response plays a decisive role in governing overall stability [1,2,3]. Dynamic disturbances, such as earthquakes, blasts, and impacts exert transient loads on rock masses, thereby generating stress waves that propagate through joints and faults. Upon encountering these discontinuities, stress waves undergo complex interaction process, including reflection, transmission, and mode conversion occur [2,3,4,5,6,7]. These processes not only redistribute wave energy and modify propagation characteristics, but can also promote the accumulation and evolution of internal damage within the rock mass, potentially leading to progressive degradation or catastrophic engineering failure [8,9,10]. Consequently, a systematic understanding of stress wave propagation across rock joints is of both fundamental scientific significance and substantial engineering relevance, providing critical insights for improving blast- and earthquake-resistant design and for reliable assessment of geological hazard risks.

Natural rock joints, often formed through long-term tectonic activity and weathering processes, are frequently infilled with weak materials such as quartz or clay, forming what are known as “filled joints” [11,12,13,14]. Compared with fresh, unfilled joints, filled joints are more widespread in nature rock masses, and their mechanical properties play a decisive role in controlling the overall dynamic response of rock masses [3,5,15,16]. In analyzing how stress waves propagate across such filled joints, the displacement discontinuity method (DDM) has been widely employed as an effective analytical framework [3,17,18,19,20,21]. This method is based on the fundamental assumption that stress remains continuous across the filled joint, while displacement exhibits a discontinuity [19,22]. By combining DDM with appropriate joint constitutive models, researchers have systematically revealed the physical mechanisms governing wave reflection, transmission, and energy dissipation at joints [3,17,18,19,20,21,23]. Under low-amplitude stress waves, filled joints typically exhibit linear deformation behavior, making linear constitutive models commonly used in such scenarios. Both numerical simulations and laboratory experiments have demonstrated that joints can induce pronounced amplitude attenuation and frequency-dependent modifications of stress waves [2,3,5,6,17,24]. However, as the amplitude of the incident wave increases, joint deformation becomes strongly nonlinear, with phenomena such as nonlinear compression hardening becoming increasingly significant. To capture this high-stress response, nonlinear joint constitutive models, most notably the Bandis–Barton model, have been introduced and widely applied [3,5,21,25,26,27,28]. Further studies have revealed that as the incident wave amplitude increases, the wave attenuation capability of the joint decreases, demonstrating a distinct amplitude-dependent behavior [27,29].

In addition, filled joints in natural rock masses are often partially or fully water-bearing, and their dynamic response is significantly influenced by the presence of pore water [30]. Under transient dynamic loads such as blasting, the water within saturated filled joints cannot dissipate or drain rapidly, effectively forming an incompressible fluid layer. This layer not only carries part of the external load, but also significantly enhances the effective normal stiffness of the filled joint, thereby influencing its displacement discontinuity behavior under both compressive and shear waves [31,32,33,34]. Meanwhile, the viscous coupling effect induced by the interstitial fluid between joint walls further affects the efficiency of wave energy transmission [25,35,36,37]. As a result, filled joints subjected to dynamic loading typically exhibit pronounced viscoelastic mechanical behavior. Notably, even for dry joints, viscoelastic constitutive descriptions have been shown to provide superior predictive capability compared with purely elastic models in simulating seismic wave propagation [38]. Based on classical viscoelastic representations, including the Kelvin, Maxwell, and Burgers models, numerous researchers have systematically investigated the influence of joint viscosity on wave propagation [2,25,36,39,40]. These studies indicate that viscous stiffness plays a dual and competing role: on the one hand, it enhances energy dissipation and reduces wave transmissivity; on the other hand, it increases the effective joint stiffness, which can facilitate wave transmission under certain conditions [2]. Collectively, these findings highlight the pivotal role of viscosity in governing the dynamic response of filled rock joints.

However, most existing studies treat nonlinear compression-hardening behavior and viscous effects as independent mechanisms. While some studies address nonlinear compressive stiffness evolution from compressive hardening, others examine viscously induced wave attenuation and frequency response in isolation. Such a decoupled treatment is insufficient to capture the coupled amplitude-dependent stiffening and rate-dependent energy dissipation that characterize filled joints subjected to high-amplitude and high-loading-rate dynamic disturbances, such as blasting and strong seismic events [37,41,42]. Consequently, the applicability of existing models remains limited in these complex dynamic regimes, and they fail to resolve the competition and synergy between nonlinearity and viscosity in stress wave propagation [2,35]. To bridge this gap, this study develops a joint constitutive model that integrally couples nonlinear compressive hardening with viscoelastic dissipation. Our objective is to systematically investigate compressive stress wave propagation across nonlinear viscoelastic joint interfaces, thereby providing a more robust and physically grounded theoretical framework for analyzing rock mass dynamics and preventing engineering hazards under high dynamic loads.

2. Methods

2.1. Problem Description

We investigate the propagation of a normally incident compressive stress wave through a rock mass containing a single filled joint. The joint is idealized as a thin, weak layer of granular material, as schematically illustrated in Figure 1, which undergoes dynamic compression upon wave impingement. Since the thickness of the filled layer is significantly smaller than the wavelength of the incident stress wave, we utilize the displacement discontinuity method (DDM) [3,16,17,18,19,20,21,23,24,43], postulating continuous stress but discontinuous displacement across the joint interface. To capture the coexisting nonlinear compression-hardening behavior and viscous dissipation of the filled joint, a nonlinear Maxwell-type constitutive model is introduced, in which a nonlinear spring element and a dashpot are connected in series. This formulation enables simultaneous representation of amplitude-dependent stiffness evolution and rate-dependent dissipation mechanisms inherent to filled joints under dynamic loading. As a result, the interaction between the incident stress wave and the joint interface gives rise to complex energy redistribution, manifested through wave reflection, transmission, and attenuation. The objective of this study is to derive the governing theoretical equations describing this coupled dynamic process, thereby quantitatively resolving the stress wave propagation across nonlinear viscoelastic joint interfaces and systematically elucidating the underlying physical mechanisms controlling wave–joint interactions in such complex structural systems.

2.2. Modeling Wave Propagation Across a Filled Joint Using the Method of Characteristics

The Method of Characteristics (MC) provides a framework for modeling one-dimensional stress wave propagation. When combined with appropriate constitutive descriptions of joint interfaces, this method has been widely extended to analyze the interaction between stress waves and rock joints [3,5,12,18,21,43]. As shown in Figure 2, the specific problem involves a normally incident compressive stress wave, originating at the position $x=0$ and time $t=0$, impinging on a nonlinear viscoelastic filled joint at $x=x_1$ in the $x-t$ plane. The solution is constructed using the left- and right-running characteristic lines, leading to the governing equations given by Cai and Zhao [17]:

$$z_p{v}_n\left(x_1+d_n,t\right)-{\sigma }_n\left(x_1+d_n,t\right)=0$$

(1)

$$v_n\left(x_1,t\right)+v_n\left(x_1+d_n,t\right)=2v_n(0,t-x_1/V_p)$$

(2)

Here, $z_p=\rho V_p$ is the compressive wave impedance of the rock. The parameters $V_p$, $\rho$, $v_n,d_n$, and ${\sigma }_n$ represent the rock P-wave velocity, rock density, particle velocity, compressive deformation (closure) of the filled joint, and compressive stress, respectively.

According to the DDM [17,18,24], the governing equations are given by:

$${\sigma }_n\left(x_1,t\right)={\sigma }_n\left(x_1+d_n,t\right)={\sigma }_n$$

(3)

$$u_n\left(x_1,t\right)-u_n\left(x_1+d_n,t\right)=d_n$$

(4)

where $u_n$ denotes the compressive displacement.

In the nonlinear Maxwell viscoelastic model, a spring and a dashpot are connected in series. Consequently, they are subjected to the same stress but experience different displacements (or deformations). The total displacement of the model, corresponding to the compressive displacement of the filled joint, equals the sum of the displacements of the two components and can be written as:

$$d_n=d_n^s+d_n^d$$

(5)

In this expression, $d_n^s$ and $d_n^d$ denote the compressive displacements of the nonlinear spring and the dashpot, respectively.

Differentiating both sides of Equation (5) with respect to time yields:

$$\dot{d_n}=\dot{d_n^s}+\dot{d_n^d}$$

(6)

The compression-hardening behavior of the nonlinear spring is characterized by the Bandis–Barton model [11]:

$$d_n^s={\displaystyle \frac{{\sigma }_n}{k_{ni}+{\sigma }_n/d_{nmax}}}$$

(7)

Here, $k_{ni}$ and $d_{nmax}$ are the initial normal stiffness and the maximum allowable closure, respectively.

Differentiating both sides of Equation (7) with respect to time yields:

$$\dot{d_n^s}={\displaystyle \frac{k_{ni}{d}_{nmax}}{{d_{nmax}(k_{ni}+{\sigma }_n/d_{nmax})}^2}}{\displaystyle \frac{\mathsf{\partial }{\sigma }_n}{\mathsf{\partial }t}}$$

(8)

Differentiating the displacement of the dashpot with respect to time yields:

$$\dot{d_n^d}={\displaystyle \frac{{\sigma }_n}{\eta }}$$

(9)

where $\eta$ denotes the viscous stiffness of the dashpot.

Combining Equations (8) and (9) with Equation (6), we obtain:

$$\dot{d_n}={\displaystyle \frac{k_{ni}}{{(k_{ni}+{\sigma }_n/d_{nmax})}^2}}{\displaystyle \frac{\mathsf{\partial }{\sigma }_n}{\mathsf{\partial }t}}+{\displaystyle \frac{{\sigma }_n}{\eta }}$$

(10)

Substituting Equations (1) and (2) into Equation (10) yields:

$$\begin{gathered} 2v_n(0,t-x_1/V_p)-{(2+z_p/\eta )v}_n\left(x_1+d_n,t\right) \\ ={\displaystyle \frac{k_{ni}{z}_p}{{(k_{ni}+z_p{v}_n\left(x_1+d_n,t\right)/d_{nmax})}^2}}{\displaystyle \frac{\mathsf{\partial }{v}_n\left(x_1+d_n,t\right)}{\mathsf{\partial }t}} \end{gathered}$$

(11)

Equation (11) can be algebraically manipulated to obtain:

$${\displaystyle \frac{\mathsf{\partial }{v}_n\left(x_1+d_n,t\right)}{\mathsf{\partial }t}}={\displaystyle \frac{2v_n\left(0,t-x_1/V_p\right)-{(2+z_p/\eta )v}_n\left(x_1+d_n,t\right)}{\frac{k_{ni}{z}_p{d_{nmax}}^2}{{(k_{ni}{d}_{nmax}+z_p{v}_n\left(x_1+d_n,t\right))}^2}}}$$

(12)

For convenience, we denote $v_n(0,t-x_1/V_p)$ as $v_n^I\left(t\right)$, where the superscript $I$ signifies the incident wave, and express the partial differential equation in its incremental form:

$${\displaystyle \frac{v_n\left(x_1+d_n,t_{j+1}\right)-v_n\left(x_1+d_n,t_j\right)}{\Delta t}}={\displaystyle \frac{2v_n^I\left(t\right)-{(2+z_p/\eta )v}_n\left(x_1+d_n,t_j\right)}{\frac{k_{ni}{z}_p{d_{nmax}}^2}{{(k_{ni}{d}_{nmax}+z_p{v}_n\left(x_1+d_n,t_j\right))}^2}}}$$

(13)

In this formulation, $\mathsf{\Delta }t$ is the time-step size.

By rearranging Equation (12), we obtain:

$$v_n\left(x_1+d_n,t_{j+1}\right)=v_n\left(x_1+d_n,t_j\right)+{\displaystyle \frac{2v_n^I\left(t_j\right)-{(2+z_p/\eta )v}_n\left(x_1+d_n,t_j\right)}{\frac{k_{ni}{z}_p{d_{nmax}}^2}{{(k_{ni}{d}_{nmax}+z_p{v}_n\left(x_1+d_n,t_j\right))}^2}}}\Delta t$$

(14)

Based on Equation (14), the transmitted wave $v_n(x_1+d_n,t_{j+1})$ across the filled joint at time $t_{j+1}$ can be recursively determined from the incident wave $v_n^I\left(t_j\right)$ and the transmitted wave $v_n\left(x_1+d_n,t_j\right)$ at time $t_j$. Therefore, once the initial conditions are specified, the transmitted wave corresponding to any incident wave time history can be solved through an iterative procedure. For convenience, we define a non-dimensional viscous stiffness ${\eta }^{\prime }=\eta /z_p$ for subsequent parametric analysis.

To facilitate the discussion, we denote the transmitted wave particle velocity $v_n(x_1+d_n,t)$ by $v_n^T(x_1+d_n,t)$, with the superscript $T$ denoting “transmitted”. The transmission coefficient $T_{coef}$ is then defined as the ratio of the absolute amplitude of the transmitted wave to that of the incident wave, given by:

$$T_{coef}={\displaystyle \frac{\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|v_n^T\left(x_1,t\right)\right|\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|v_n^I\left(t\right)\right|\right)}}$$

(15)

According to the stress continuity condition across the filled joint, the stress at the front of the joint, which is the sum of the incident stress wave ${\sigma }_n^I\left(x_1,t\right)$ and the reflected stress wave ${\sigma }_n^R(x_1,t)$, must equal the stress at the back of the joint, namely the transmitted stress wave ${\sigma }_n^T(x_1+d_n,t)$. This relationship is expressed as:

$${\sigma }_n^R\left(x_1,t\right)+{\sigma }_n^I\left(x_1,t\right)={\sigma }_n^T\left(x_1+d_n,t\right)$$

(16)

From Equation (1), it follows that:

$${z_{p}v}_n^R\left(x_1,t\right)+{z_{p}v}_n^I\left(t\right)={z_{p}v}_n^T\left(x_1+d_n,t\right)$$

(17)

Thus, the corresponding reflected wave can be determined:

$$v_n^R\left(x_1,t\right)=v_n^T\left(x_1+d_n,t\right)-v_n^I\left(t\right)$$

(18)

The reflection coefficient $R_{\mathrm{coef}}$ is defined as the ratio of the absolute amplitudes of the reflected wave to the incident wave:

$$R_{coef}={\displaystyle \frac{\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|v_n^R\left(x_1,t\right)\right|\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|v_n^I\left(t\right)\right|\right)}}$$

(19)

Due to the dissipative nature of the filled joint, which causes wave energy attenuation, a primary objective of this study is to analyze the corresponding energy loss.

The incident wave energy $E_I$ is defined as the time integral of the product of the stress ${\sigma }_n^I\left(t\right)$ and strain ${\epsilon }_n^I\left(t\right)$. For computational convenience, the following discretized form is employed:

$$E_I={\int }_0^t{\sigma }_n^I\left(t\right){\epsilon }_n^I\left(t\right)dt\approx \sum _{j=0}^{j=t}{\sigma }_n^I\left(t\right){\epsilon }_n^I\left(t\right)∆t$$

(20)

According to Zhao and Cai [28], the strain ${\epsilon }_n^I\left(t\right)$ and the particle velocity $v_n^I\left(t\right)$ are related by:

$${\epsilon }_n^I\left(t\right)=v_n^I\left(t\right)/V_p$$

(21)

Hence, Equation (20) can be expressed in the form of:

$$E_I\approx \sum _{j=0}^{j=t}{z}_p{\left(v_n^I\left(t\right)\right)}^2∆t/V_p$$

(22)

Likewise, we can write the reflected and transmitted wave energies, $E_R$ and $E_T$, as:

$$E_R\approx \sum _{j=0}^{j=t}{z}_p{\left(v_n^R\left(x_1,t\right)\right)}^2∆t/V_p$$

(23)

$$E_T\approx \sum _{j=0}^{j=t}{z}_p{\left(v_n^T\left(x_1+d_n,t\right)\right)}^2∆t/V_p$$

(24)

We define the energy attenuation coefficient $\delta$ as:

$$\delta =1-{\displaystyle \frac{E_R+E_T}{E_I}}=1-{\displaystyle \frac{\sum _{j=0}^{j=t}{{(v}_n^R\left(x_1,t\right))}^2∆t+\sum _{j=0}^{j=t}{{(v}_n^T\left(x_1,t\right))}^2∆t}{\sum _{j=0}^{j=t}{{(v}_n^I\left(t\right))}^2∆t}}$$

(25)

To simplify the analysis, a half-cycle sinusoidal pulse is adopted for the incident compressive wave, which takes the form of:

$$v_n^I\left(t\right)=Asin\left(2\pi ft\right)\dots 0\le t\le 1/\left(2f\right)$$

(26)

where $A$ and $f$ are the amplitude and frequency, respectively.

Figure 3 presents the flowchart of the proposed computational procedure (Supplementary Materials). This flowchart illustrates a complete workflow for simulating wave propagation through a nonlinear medium. The procedure begins with the initialization of medium, wave, and temporal parameters, followed by the generation of a discrete time vector and the construction of an incident wave in the form of a half-cycle sinusoidal pulse within a specified time interval. Next, the transmitted wave is computed iteratively using a numerical integration scheme that accounts for nonlinear material effects via two denominator terms and corresponding update coefficients, while enforcing numerical stability and non-negativity constraints. Finally, the reflected wave is obtained by subtracting the incident wave from the transmitted wave, completing the simulation and providing the resulting waveforms for subsequent analysis of wave interactions and energy dissipation.

3. Results

This section begins with a validation of the proposed nonlinear–viscous coupled model through a comparative analysis. Specifically, under identical experimental conditions and incident wave inputs, the predictions of the proposed model are compared with those obtained from the conventional uncoupled Bandis–Barton model as well as with modified split Hopkinson pressure bar (SHPB) experimental results. This comparison demonstrates the effectiveness and advantages of the proposed coupled model in reproducing the key features of stress wave transmission and reflection through filled joints. Following the model validation, the influence of incident wave characteristics and joint constitutive parameters on wave propagation through viscoelastic filled joints is systematically investigated. Incident wave amplitude and frequency are employed to characterize the wave field, whereas the initial normal stiffness, maximum allowable closure, and viscous parameter describe the constitutive behavior of the filled joint. These parameters are varied individually to clarify their respective effects on transmitted and reflected wave responses as well as on energy attenuation.

3.1. Model Validation

To evaluate the applicability and accuracy of the proposed analytical model, a modified split Hopkinson pressure bar (SHPB) experiment was conducted for model validation. The experiment aimed to characterize stress wave propagation through an artificially filled joint under dynamic loading. A schematic diagram of the modified SHPB apparatus is shown in Figure 4. The experimental setup mainly consists of a gas gun, a striker bar, an incident bar, a transmission bar, two pairs of strain gauges, and an oscilloscope. All bars were fabricated from high-quality steel with a density of 7980 kg/m³ and a longitudinal wave velocity of 5058 m/s. The striker, incident, and transmission bars have a uniform diameter of 25.4 mm, with corresponding lengths of 200 mm, 2000 mm, and 1500 mm, respectively. Two pairs of strain gauges were mounted on the bars, one pair on the incident bar at a distance of 1000 mm from the impact end, and the other on the transmission bar located 1100 mm from the free end.

Figure 5 presents the incident, transmitted, and reflected waveforms recorded in the SHPB test conducted under an impact gas pressure of 193.05 kPa, corresponding to an artificially filled joint with a thickness of 5.01 mm. To assess the feasibility and accuracy of the proposed coupled constitutive model, the experimental waveforms were forward fitted using both the nonlinear–viscous coupled model (shown by blue lines in Figure 5) and the conventional Bandis–Barton model (shown by green lines in Figure 5). For the coupled model, the parameters are $k_{ni}$ = 8.9 × 10¹¹ Pa/m, $d_{nmax}$ = 0.5 mm, and ${\eta }^{\prime }$ = 1.8, whereas the conventional Bandis–Barton model adopts 1.8 × 10¹¹ Pa/m and $d_{nmax}$ = 1.5 mm. As shown in Figure 5, for the transmitted wave, the coupled model (blue lines) reproduces both the overall waveform shape and peak amplitude with good agreement to the experimental data. In contrast, the conventional Bandis–Barton model (green lines) shows noticeable discrepancies in the amplitude and temporal evolution of the transmitted response, particularly during the rising and descending stages of the waveform. These deviations indicate limitations of the Bandis–Barton model in capturing the effective stiffness evolution of the filled joint during dynamic loading. For the reflected wave, the coupled model (blue lines) provides a closer match to the experimental waveform in both amplitude and phase. In contrast, the conventional Bandis–Barton model (green lines) overestimates the reflection amplitude and exhibits a noticeable phase shift relative to the experimental record. The improved agreement achieved by the coupled model for both transmitted and reflected waves demonstrates its enhanced capability in describing stress wave partitioning at the joint interface. Overall, the comparison in Figure 5 shows that incorporating nonlinear stiffness and viscous effects enables the proposed model (blue lines) to achieve a more consistent and simultaneous fit to both transmitted and reflected waveforms, while the conventional Bandis–Barton model (green lines) struggles to reproduce these features.

3.2. Waveform

Figure 6a presents the transmitted and reflected waves generated by incident waves of different amplitudes passing through a viscoelastic filled joint. It can be observed that both the transmitted and reflected waves exhibit longer durations than the incident wave, indicating waveform broadening and a significant reduction in their dominant frequency compared to the incident wave. The transmitted wave remains compressive, while the reflected wave initially manifests as tensile before transitioning to compressive. With increasing incident wave amplitude, the amplitude of the transmitted wave increases and its duration shortens; the amplitude of the reflected wave also increases, but its duration shows no clear variation. Notably, when the incident wave amplitude increases to 1.0 m/s and 1.5 m/s, pronounced distortion occurs during the unloading phase of the reflected waveform. This behavior is attributed to the nonlinear compression-hardening of the joint, higher incident amplitudes induce stronger stiffening, accelerating wave transmission and modifying the reflection waveform.

Figure 6b shows the transmitted and reflected waves resulting from an incident wave passing through viscoelastic filled joints with different initial compressive stiffnesses. Compared to Figure 6a, the transmitted and reflected waveforms exhibit more symmetrical and uniform variations, with no distortion observed in the reflected wave. As the initial compressive stiffness increases, the amplitude of the transmitted wave gradually increases, and its duration shortens. However, unlike the case in Figure 6a, both the amplitude and duration of the reflected wave decrease accordingly in this scenario. The trend reflects that higher initial stiffness enhances the joint’s resistance to deformation, promoting wave energy transmission and reducing reflection.

Figure 6c illustrates the transmitted and reflected waves generated by an incident wave passing through viscoelastic filled joints with different maximum allowable closures. For the transmitted wave, its amplitude decreases as the maximum allowable closure increases, while its duration lengthens. For the reflected wave, waveform distortion occurs when the maximum allowable closure is 0.2 mm; as the maximum allowable closure further increases, the distortion disappears. The amplitude of the reflected wave increases sharply at first and then the rate of increase slows, while its duration shows little change. This is because a smaller allowable closure limits joint compression, leading to earlier stiffening and increased reflection, whereas larger closure allows more deformation before stiffening occurs.

Figure 6d displays the transmitted and reflected waves produced by an incident wave passing through viscoelastic filled joints with different viscous stiffnesses. Compared to the other parameters examined in Figure 6a–c, the influence of viscous stiffness on the transmitted and reflected waves is relatively minor. As the viscous stiffness increases, the amplitude of the transmitted wave rises significantly initially and then the change levels off; the reflected wave, however, remains largely unaffected by variations in viscous stiffness. Furthermore, the durations of both the transmitted and reflected waves show no significant change with increasing viscous stiffness. This pattern highlights the dual role of viscous damping, it can slightly increase effective stiffness, enhancing transmission, but also dissipates energy, which primarily affects amplitude rather than waveform shape.

3.3. Reflection and Transmission Coefficients

To further quantify the sensitivity of wave propagation to constitutive parameters under different wave-filed conditions, reflection and transmission coefficients are systematically analyzed for varying both the incident wave characteristics and the constitutive parameters of the filled joint. Figure 7 illustrates the variation in transmission and reflection coefficients with the amplitude of the incident wave. As the incident wave amplitude increases, the transmission coefficient rises sharply and the reflection coefficient decreases dramatically in the initial stage. With further increase in incident amplitude, the trend levels off: the transmission coefficient stabilizes around 0.75, while the reflection coefficient continues to decrease slowly, following an overall negative exponential trend. These results indicate that at low incident amplitudes, the joint is relatively compliant, reflecting more energy and making the transmission and reflection coefficients highly sensitive to amplitude variations, as the amplitude increases, nonlinear hardening stiffness the joint, enhancing transmission and reducing reflection, so that the amplitude dependency gradually weakens at higher amplitudes.

Figure 8 illustrates the variation in transmission and reflection coefficients with the frequency of the incident wave. In the initial low-frequency range, both coefficients remain stable without significant changes. As the frequency increases, the reflection coefficient begins to increase nonlinearly, with the slope of its rising curve gradually decreasing, indicating a saturation trend in the growth. Meanwhile, the transmission coefficient starts to decrease nonlinearly once the reflection coefficient begins to rise. The absolute value of the slope of the transmission curve first increases and then stabilizes, suggesting that the transmission capacity undergoes an initial rapid decline before transitioning to a more gradual attenuation process. This behavior arises because at higher frequencies, the viscous element cannot fully respond, increasing effective compliance and enhancing reflection.

Figure 9 demonstrates the relationship between the initial normal stiffness and the wave coefficients. This parameter exhibits a predominant influence on the reflection coefficient, which shows a sharp decrease in response to increasing stiffness, following a decay pattern akin to a negative exponential function with rapid initial decline followed by gradual stabilization. In contrast, the transmission coefficient forms an essentially horizontal plateau, clearly indicating its high degree of independence from this specific stiffness parameter.

Figure 10 shows the variation in transmission and reflection coefficients with the maximum allowable closure of the filled joint. In the initial stage, when the maximum allowable closure is relatively small, both coefficients remain stable without significant changes. As the closure gradually increases, the reflection coefficient begins to rise nonlinearly, with the slope of its ascending curve gradually decreasing as the closure increases, reflecting a trend toward saturation. Meanwhile, the transmission coefficient exhibits a nonlinear decrease, though the overall attenuation remains relatively gradual. Smaller maximum allowable closure leads to earlier stiffening of the joint, promoting energy transmission, whereas larger closure delays stiffening, which can reduce transmission and increase reflection.

Figure 11 shows the variation in the transmission and reflection coefficients with the viscous stiffness of the filled joint. As the viscous stiffness increases, the reflection coefficient decreases nonlinearly, with a rapid initial decline that gradually levels off. In contrast, the transmission coefficient increases nonlinearly, characterized by a sharp rise in the initial stage followed by a progressively narrowing rate of increase. This is consistent with viscous dissipation reducing energy loss in transmission at moderate stiffness while also limiting reflection through increased effective damping.

3.4. Energy Dissipation Coefficient

Compared with waveform characteristics and wave coefficients, the energy attenuation coefficient provides an integrated measure of the sensitivity of joint response to constitutive parameters across different wave-field conditions. Figure 12a–c systematically illustrates the nonlinear response of the energy attenuation coefficient to variations in incident wave amplitude. Overall, all three sets of curves exhibit an evolutionary trend characterized by an initial rapid increase, followed by a gradually slowing growth rate, and eventual stabilization. This trend reflects the combined effects of nonlinear compression and viscous damping: energy dissipation is initially dominated by nonlinear stiffening, and then viscous relaxation becomes more influential as amplitude increases.

Figure 12a compares the influence of different initial compressive stiffness values. In the low amplitude regime, the stiffness effect is particularly prominent, where higher initial compressive stiffness corresponds to a greater attenuation coefficient. This indicates that joint initial compressive stiffness dominates wave energy transmission under weak disturbances. As the wave amplitude increases, the gap between the curves gradually narrows. When the incident wave amplitude exceeds 6 m/s, the curves nearly converge, demonstrating that the influence of initial compressive stiffness is substantially diminished under high energy incident conditions, and the structural response transitions from linear to nonlinear. Higher stiffness leads to faster joint hardening, increasing transmitted energy and reducing reflection, which affects energy dissipation trends.

Figure 12b focuses on the role of the maximum allowable closure. At lower wave amplitudes, the influence of this parameter is relatively limited, with larger closure corresponding to a slightly lower attenuation coefficient. As the wave amplitude rises, the differences between curves corresponding to different closure values become most pronounced in the medium amplitude range, indicating that the filled joint’s compressive potential is fully activated at this stage, making closure a key factor controlling energy dissipation. In the high amplitude regime, the curves converge again, suggesting that the filled joint closure approaches its limit and its influence diminishes. This behavior arises because earlier stiffening with smaller closure enhances energy transmission, while larger closure delays stiffening, reducing transmitted energy and increasing reflected energy.

Figure 12c reveals the moderating role of viscous stiffness on energy attenuation. In the low amplitude range, the influence of viscous stiffness is weak, where higher values correspond to a slightly lower attenuation coefficient. As the wave amplitude increases, the differences between curves corresponding to different viscous stiffness values gradually widen and eventually stabilize in the high amplitude region. This indicates that the viscous dissipation mechanism is significantly activated under strong dynamic excitation, becoming one of the primary factors affecting energy attenuation.

Figure 13a–c systematically illustrates the nonlinear response of the energy attenuation coefficient to variations in incident wave frequency. Overall, all three sets of curves exhibit a general decreasing trend with increasing frequency, though the specific evolution characteristics are notably influenced by different joint parameters.

Figure 13a reflects the influence of initial compressive stiffness on the energy attenuation coefficient. As the frequency increases, the attenuation coefficient first decreases gradually and then drops rapidly. In the low-frequency stage, the differences between curves with different stiffness values are relatively small. However, as the frequency rises, the influence of stiffness becomes increasingly significant, manifesting as higher initial compressive stiffness corresponding to greater energy attenuation coefficients, with the gap between the curves progressively widening. This indicates that initial compressive stiffness primarily affects energy attenuation at higher frequencies, where the joint deformation is more sensitive to stiffness variations.

Figure 13b demonstrates the effect of the maximum allowable closure. At low frequencies, the influence of this parameter on the attenuation coefficient is limited. As the frequency increases, the differences between curves corresponding to different closure values gradually expand, reach a maximum, and then narrow again. Furthermore, with increasing closure, the attenuation coefficient gradually decreases, and the curve morphology transitions from a convex shape to a concave shape. This suggests that larger closures delay the onset of stiffening, reducing the joint’s ability to attenuate energy at intermediate frequencies.

Figure 13c reveals the moderating mechanism of viscous stiffness on energy attenuation. As the frequency increases, the attenuation coefficient shows an evolution process characterized by an initial sharp decline followed by gradual stabilization, overall following a negative exponential decay pattern. In the low-frequency stage, the influence of viscous stiffness is relatively pronounced, with higher values corresponding to smaller attenuation coefficients. As the frequency further increases, the differences between curves corresponding to different viscous stiffness values gradually diminish and eventually converge. This reflects that viscous damping mainly governs energy attenuation at low to moderate frequencies, while its effect becomes limited at high frequencies due to the short interaction time per wave cycle.

4. Discussion

Filled joints in rock masses, as typical geological discontinuities, play a critical controlling role in stress wave propagation. Their unique mechanical behavior not only governs the reflection and transmission characteristics of stress waves but also profoundly influences the dissipation and transmission efficiency of wave energy. When a compressive stress wave propagates to a filled joint, the stress remains continuous across the interface, while the displacement exhibits a discontinuous jump. This displacement discontinuity leads to a complex decomposition process and energy redistribution of the stress wave. Therefore, gaining an in-depth understanding of the coupling relationship between the filled joint’s compressive deformation mechanism and stress wave parameters holds significant theoretical value for rock dynamics analysis.

Under compressive loading, filled joints exhibit significant nonlinear hardening characteristics, which can be accurately described by the Bandis–Barton model. This model is governed by two key parameters: the initial compressive stiffness and the maximum allowable closure. Research shows that a higher initial compressive stiffness and a smaller maximum allowable closure result in a more pronounced stiffness hardening effect during compression. This nonlinear stiffness characteristic significantly influences stress wave propagation: when the incident wave amplitude increases, the filled joint rapidly hardens under high pressure, leading to a significant increase in stiffness, which strengthens the transmission of wave energy, manifesting as reduced reflection and enhanced transmission; conversely, when the incident wave frequency increases, the filled joint cannot fully close under rapid loading, exhibiting higher equivalent compliance, which leads to enhanced reflection and reduced transmission.

It is noteworthy that filled joints in natural rock masses not only exhibit nonlinear compression-hardening behavior but also demonstrate significant viscous characteristics. By connecting a spring representing nonlinear compressive hardening in series with a dashpot representing viscous damping, a nonlinear Maxwell viscoelastic model can be constructed. In this model, the spring and the dashpot experience the same stress but exhibit different displacement responses, and together they determine the overall mechanical behavior of the filled joint based on their respective stiffness characteristics. The introduction of viscous stiffness has a dual effect: on one hand, it effectively increases the overall stiffness of the filled joint system, helping to suppress stress wave reflection and enhance transmission capacity; on the other hand, the viscous dissipation mechanism converts part of the wave energy into heat, thereby reducing the amplitudes of both the reflected and transmitted waves.

In a composite model that considers both nonlinear compression-hardening and viscous behavior, the filled joint’s response mechanism to stress waves becomes more complex. Multiple parameters, including the amplitude and frequency of the incident wave, as well as the filled joint’s initial compressive stiffness, maximum allowable closure, and viscous stiffness, collectively regulate the wave propagation process. Specifically, under high-amplitude stress waves, the nonlinear spring undergoes significant compression hardening, causing a sharp increase in the filled joint’s overall stiffness. If coupled with high viscous stiffness, the system exhibits stronger transmission capacity. Under high-frequency incident waves, however, the response lag of the viscous element becomes more pronounced, causing the system to exhibit more obvious reflection characteristics and energy dissipation. These observations indicate that wave propagation through filled joints is governed by competing and interacting mechanisms, including nonlinear stiffness hardening, viscous dissipation, and wave-filed characteristics. The dominance of each mechanism depends on the relative scales of loading amplitude, frequency, and joint constitutive properties, rather than on any single parameter alone. To quantitatively capture these competing “relative scales” and establish more general scaling laws, a future extension of this work could employ dimensional analysis (e.g., following the approach in Li et al. [44]). This would involve synthesizing the multiple governing parameters identified here into key dimensionless numbers, such as a viscoelastic contrast ratio and a nonlinear driving level, offering a unified framework to predict wave behavior across different physical scales.

Further analysis shows that the reflection coefficient, transmission coefficient, and energy dissipation coefficient are direct manifestations of the coupling effects of these parameters. A higher initial compressive stiffness weakens the filled joint’s hindering effect on stress waves, leading to a corresponding increase in the transmission coefficient; a smaller maximum allowable closure causes the filled joint to enter a high-stiffness state earlier, favoring the propagation of high-frequency wave components; while an increase in viscous stiffness, although improving transmission efficiency under specific conditions, inevitably accompanies enhanced energy dissipation, thereby significantly altering the energy distribution characteristics of the wave field. From a global perspective, these coefficients can be regarded as macroscopic indicators of the coupled joint–wave system response, reflecting the combined influence of nonlinear deformation capacity, viscous relaxation behavior, and wave-field characteristics.

In addition to amplitude and frequency effects, the propagation behavior of stress waves across filled joints is also influenced by the shape of the incident waveform. In the present study, half-sinusoidal pulses were primarily adopted to provide smooth and well-controlled excitation, allowing the fundamental coupling mechanisms between nonlinear compression hardening and viscous dissipation to be clearly identified. However, realistic dynamic events such as blasting, impact loading, and near-field seismic waves are often characterized by highly asymmetric waveforms with peak values occurring at the early stage of loading. To qualitatively examine the influence of waveform characteristics, an illustrative example using asymmetric triangular incident waves was further considered, in which a dimensionless peak-position parameter $\alpha$ was introduced to characterize incident-wave asymmetry. It is defined as the ratio of the loading (rising) duration to the total pulse duration, such that smaller $\alpha$ corresponds to a more rapid loading process with the wave peak arriving earlier in time. As demonstrated in Figure 14, incident waves with earlier-arriving peaks (i.e., smaller $\alpha$) result in increased reflection and reduced transmission across the filled joint. This behavior can be attributed to the more abrupt loading process associated with smaller $\alpha$ values. For a fixed peak amplitude, rapid loading limits the development of nonlinear compression hardening at the joint interface, leading to a stronger transient impedance mismatch. Meanwhile, the higher closure rate enhances viscous resistance, which preferentially attenuates rapidly varying wave components. These results indicate that waveform characteristics can significantly influence the relative contributions of nonlinear stiffness evolution and viscous dissipation during wave–joint interaction. Although a comprehensive parametric investigation of waveform effects is beyond the scope of the present study, this triangular-wave example demonstrates that the proposed nonlinear viscoelastic model is capable of capturing waveform-dependent behaviors. A systematic study of shock-type or strongly asymmetric incident waves will be pursued in future work to further extend the applicability of the proposed framework.

In summary, the coupling between the nonlinear compression-hardening characteristics and viscous behavior of filled joints jointly governs the propagation laws of stress waves in rock masses. An in-depth understanding of the coupling mechanism under the series model not only reveals the intrinsic physical nature of wave propagation but also provides a solid theoretical foundation for the dynamic stability assessment of rock engineering and the design of wave control measures. Future research should focus on a more systematic evaluation of the coupled effects of wave-field characteristics and joint constitutive parameters, thereby promoting more reliable application of this theoretical model in engineering practice.

5. Conclusions

The propagation behavior of compressive stress waves across filled joints is controlled by the coupled interplay between nonlinear compression-hardening and viscosity. Theoretical analysis demonstrates that wave transmission capacity increases with incident amplitude but decreases with frequency, whereas the reflection response exhibits the opposite trend. Parametric investigations further indicate that an increasing initial compressive stiffness leads to a nonlinear reduction in the reflection coefficient while exerting only a marginal influence on transmission. In contrast, a larger maximum allowable closure strongly suppresses wave transmission while significantly enhancing reflection, whereas greater viscous stiffness effectively promotes transmission. Energy attenuation is found to increase rapidly with amplitude before reaching a saturation regime. In this process, the initial compressive stiffness dominates the response at low amplitudes, the maximum allowable closure plays a critical role at intermediate amplitudes, and viscous stiffness remains influential over the entire amplitude range. With increasing frequency, attenuation decreases in a nonlinear manner, accompanied by an enhanced contribution of compression-hardening parameters at high frequencies, while viscous effects prevail at low frequencies. These findings provide critical insights into wave propagation mechanisms through filled joints, establishing a theoretical basis for rock mass stability assessment and wavefield interpretation.