⚡ electrical engineering

Delayed-Memory Dynamics of Rock-Mass Instability During TBM Metro Tunnel Excavation

This study develops a reduced-order nonlinear model incorporating delayed memory and friction dynamics to explain how TBM advance rates and cumulative rock weakening govern the transition between stable, delayed clustered, and impulsive instability regimes during metro tunnel excavation in jointed rock masses.

Original authors: Srđan Kostić

Published 2026-08-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Srđan Kostić

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep beneath the city, where the earth is often fractured and faulted, a massive machine known as a tunnel boring machine, or TBM, pushes forward to carve out the paths for new metro lines. These machines are marvels of engineering, capable of cutting through rock with a steady, continuous motion. However, the ground they travel through is not a solid, uniform block. It is a complex assembly of stone blocks held together by cracks, joints, and ancient fault lines. When the machine advances, it does not simply remove material; it changes the way the surrounding rock is squeezed and stretched. This change in pressure can cause the rock to crack, slip, or even burst, sometimes in ways that are difficult to predict. For decades, engineers have relied on detailed maps of the rock and calculations of the stress at the exact moment the machine passes a specific point to ensure safety. But field observations have shown that the story is more complicated. Sometimes, the rock remains quiet as the machine passes, only to shift, crack, or release small tremors days or even weeks later. This delayed reaction suggests that the rock remembers the disturbance long after the machine has moved on, and that this memory, combined with the time it takes for stress to travel through the ground, plays a critical role in when and where instability occurs.

A researcher at the Jaroslav Černi Water Institute in Serbia has developed a new way to understand this hidden timing. Instead of trying to map every single crack in a specific tunnel, which requires immense computing power and detailed geological surveys, the researcher created a simplified, yet powerful, mathematical model. Imagine the rock mass not as a continuous solid, but as two distinct blocks of stone sitting next to each other, connected by a spring and separated by a rough surface that resists sliding. As the tunnel boring machine moves forward, it acts like a moving wave of pressure that passes over these blocks. The model tracks how the blocks move, how they rub against each other, and how they change internally over time. Crucially, the model includes two concepts that are often overlooked in standard engineering calculations: memory and delay. The "memory" represents the idea that the rock gets weaker the more it is stressed and damaged; it retains a record of the trouble it has been through, making it more likely to slip in the future. The "delay" accounts for the fact that when one part of the rock shifts, it takes a little time for that movement to affect the neighboring part, much like a ripple traveling across a pond.

The researcher ran thousands of simulations with this model, changing the speed of the machine, the strength of the rock, and the amount of time it takes for stress to travel between the blocks. The results revealed that the relationship between the machine's speed and the rock's safety is not a simple straight line. It was found that moving the machine very slowly can actually be dangerous. When the machine crawls forward, it stays in front of a specific section of rock for a long time, allowing the "memory" of the stress to build up and the rock to weaken significantly. This slow accumulation of damage can eventually lead to a sudden, delayed slip long after the machine has passed. Conversely, moving the machine very fast creates a different kind of danger. The rapid change in pressure acts like a sharp shock, triggering immediate, impulsive slips right near the machine's face. Surprisingly, the simulations identified a narrow "sweet spot" in the middle. At an intermediate speed, the machine moves fast enough to prevent the rock from accumulating too much damaging memory, but slow enough to avoid the sharp shocks of rapid advancement. In this specific window, the system remained stable, suggesting that there is an optimal pace for excavation that balances these competing risks.

One of the most striking discoveries was how the instability spreads. The model showed that the rock does not always fail in the order the machine encounters it. Often, the block of rock that the machine reaches second would become unstable before the first block, even though the machine passed the first one earlier. This happened because the second block was receiving a "double dose" of trouble: the direct pressure from the machine passing over it, and a delayed, weakened signal from the first block that had already been disturbed. This led to a specific pattern of behavior that the researcher calls "selective activation." Before the entire system collapses into a chaotic mess, the second block often starts to slip on its own. This selective slipping acts as a precursor, a warning sign that the rock mass is approaching a critical point. The model suggests that if engineers see activity starting in a downstream area while the upstream area remains quiet, it could indicate that the delayed interactions and accumulated damage are building up to a larger, coupled failure.

The study also highlighted the importance of the time lag between disturbances. The simulations showed that the delay in stress transfer could either calm the rock down or make it worse, depending on the timing. Sometimes, a delay would cause the blocks to move out of sync in a way that stabilized them; other times, the delay would cause them to push against each other at the exact wrong moment, amplifying the movement and triggering a cascade of slips. This means that the timing of the rock's reaction is just as important as the amount of force applied. The research does not claim to predict exactly when a specific tunnel will fail, nor does it replace the detailed 3D maps engineers use for design. Instead, it offers a new lens for understanding the "when" and "why" of delayed failures. It suggests that the rock's history of damage and the time it takes for stress to travel through it are the keys to unlocking the mystery of why some tunnels remain quiet while others experience sudden, delayed bursts of activity. By recognizing these patterns, particularly the early signs of selective activation in downstream zones, monitoring systems could potentially be tuned to spot these dangerous transitions before they escalate into major instability events.

Technical Summary: Delayed-Memory Dynamics of Rock-Mass Instability During TBM Metro Tunnel Excavation

Problem Statement
Tunnel boring machine (TBM) excavation in jointed and faulted rock masses often triggers complex responses, including delayed fracture activation, intermittent slip, and spatially migrating microseismic activity. These phenomena are difficult to interpret using traditional quasi-static excavation stages or standard continuum/discontinuum models, which primarily focus on spatial stress redistribution at specific time steps. While field monitoring indicates that rock mass response is controlled by loading history, accumulated damage, and time-dependent deformation, systematic exploration of long-term temporal regimes—specifically delayed activation, cumulative degradation, and transitions between stable and irregular responses—remains computationally demanding and often overlooked. The study addresses the gap in understanding how the moving mechanical perturbation of a TBM face interacts with the internal memory and delayed stress transfer of a rock mass to generate delayed instability.

Methodology
The study develops a reduced-order nonlinear dynamic model representing the TBM-rock mass system as two mechanically interacting rock domains (blocks). The formulation integrates four key mechanisms:

Moving Excavation Forcing: The advancing TBM face is modeled as a moving source of stress perturbation, defined by a spatial transfer function that creates asymmetric loading ahead of and behind the face.
Nonlinear Friction: A reduced-order stick-slip representation uses a cubic friction-velocity law to capture velocity-strengthening and velocity-weakening intervals, essential for dynamic instability.
Memory-Dependent Weakening: An internal state variable ( $\chi$ ) tracks cumulative damage and frictional degradation. This variable evolves based on mechanical activity (slip, interaction, and forcing) and decays via a relaxation term, effectively reducing the frictional resistance over time.
Delayed Interaction: A finite time delay ( $\tau$ ) is introduced in the coupling between blocks to represent lagged stress transfer, fracture propagation, or pore-pressure redistribution.

The system is treated as a set of coupled stochastic delay differential equations. Numerical simulations employ a fixed-step fourth-order Runge-Kutta scheme with Gaussian increments for stochastic heterogeneity. The analysis focuses on bifurcation diagrams, phase-space trajectories, and event statistics (velocity peaks, event counts, and delays) across a range of dimensionless parameters, including TBM advance rate, forcing amplitude, delay, memory accumulation/relaxation coefficients, and friction sensitivity.

Key Contributions
The study makes four primary contributions to the modeling of TBM excavation dynamics:

Moving-Face Reformulation: It adapts the spring-block concept from static or uniformly driven systems to a moving TBM face, capturing the transient nature of excavation-induced loading.
Internal Memory Mechanism: It introduces a memory variable to explicitly model the evolution of effective frictional resistance due to cumulative damage, allowing the system to "remember" past loading cycles.
Delayed Coupling: It incorporates finite interaction delays to describe the lag between face passage and the mechanical response of coupled rock domains.
Regime-Based Interpretation: It establishes a framework for classifying responses into stable, selective, and coupled regimes, isolating how combinations of mechanisms generate delayed instability rather than just predicting spatial stress fields.

Numerical Results and Dynamic Regimes
The simulations reveal that instability is not a monotonic function of excavation parameters but emerges from the interaction between internal weakening and external excitation. Key findings include:

Stable vs. Unstable Transitions: The system exhibits sharp transitions from stable deformation to dynamic slip. A critical finding is the existence of a selective regime, where the downstream domain activates before the upstream domain, despite the face passing the upstream domain first. This reversal is driven by the combined effect of local loading history and delayed mechanical interaction.
Non-Monotonic Advance Rate Effect: The TBM advance rate ( $v_f$ ) has a non-monotonic impact. Slow rates promote memory-dominated instability due to prolonged exposure and damage accumulation. Intermediate rates create a narrow stabilization window where exposure is too short for significant weakening but loading is not impulsive enough to trigger immediate failure. Rapid rates induce impulsive instability near the face.
Delay-Induced Phase Effects: Interaction delay ( $\tau$ ) produces alternating windows of stability and instability. Depending on the phase relation between the delayed coupling force and the block motion, delay can either amplify oscillations or suppress them.
Delayed Post-Passage Events: In near-critical scenarios (Case B1), dynamic events occur significantly after the face has passed the rock domain. This delayed activation is driven by the slow evolution of the memory variable, which continues to degrade friction even as the direct excavation forcing decays.
Parameter Sensitivity: The transition from stable to selective to coupled regimes is consistent across variations in forcing amplitude, delay, memory accumulation, relaxation, and friction sensitivity. The downstream domain consistently acts as the more sensitive indicator of system destabilization.

Significance and Claims
The paper positions this model not as a replacement for high-fidelity continuum or discontinuum analyses, but as a complementary tool to isolate the temporal mechanisms governing delayed instability. The author claims that the model provides testable hypotheses for interpreting field monitoring data, specifically:

Precursor Patterns: The "selective downstream activation" is identified as a robust precursor regime preceding fully coupled instability. Monitoring strategies should track the spatial migration of activity, not just total event counts.
Delayed Instability: The model suggests that instability zones cannot be defined solely by the instantaneous TBM face position; the rock mass behind the face remains dynamically sensitive due to memory effects.
Operational Implications: The non-monotonic relationship with advance rate implies that simply slowing down or speeding up excavation does not guarantee stability; the optimal rate depends on the balance between damage accumulation and loading rates.

The study concludes that excavation-induced instability is an emergent property of the interaction between moving forcing, delayed stress transfer, and memory-dependent weakening. The framework serves as a hypothesis-generating tool to guide future calibration against TBM operational records and microseismic monitoring data, offering a conceptual basis for understanding complex, time-dependent rock mass behavior.

Technical Summary: Delayed-Memory Dynamics of Rock-Mass Instability During TBM Metro Tunnel Excavation

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