Breakdown of the classical rupture theory and earthquake propagation in the "forbidden" super-Rayleigh range

By developing a rupture theory that incorporates slip-rate-dependent friction, this study demonstrates that earthquakes can continuously propagate through the previously "forbidden" super-Rayleigh speed range into the super-shear regime without a sharp transition, thereby challenging the classical two-dimensional rupture theory.

The Gist

Imagine the Earth's crust as two giant, rough blocks of rock pressed tightly together. When they suddenly slip past one another, it creates an earthquake. The "rupture" is the crack that races along the boundary between these blocks.

For decades, scientists believed there was a strict speed limit for how fast this crack could travel, based on a set of rules called "classical rupture theory." Here is the breakdown of what this paper discovered, using simple analogies.

The Old Rule: The "Forbidden Zone"

Imagine a highway with three speed zones:

1. Slow Lane: The crack moves slower than the speed of sound in the rock (sub-Rayleigh).
2. The Forbidden Zone: A mysterious gap between the "surface wave speed" and the "shear wave speed."
3. Fast Lane: The crack moves faster than the shear wave speed (super-shear).

The old theory said that if a crack wanted to go from the Slow Lane to the Fast Lane, it couldn't just speed up gradually. It had to hit a wall (the Forbidden Zone), stop, and then suddenly "jump" or teleport to the Fast Lane. This jump was called a "super-shear transition." The theory claimed the Forbidden Zone was impossible to cross smoothly because the physics of the rock would break down there.

The New Discovery: The "Speed Limit" Was Wrong

The authors of this paper decided to re-examine the rules of the road. They realized the old theory made a big assumption: it treated the friction between the rocks like a static, unchanging force, like a heavy blanket that doesn't care how fast you pull it.

In reality, friction is more like honey. If you pull honey slowly, it's thick and sticky. If you pull it very fast, it behaves differently; it gets thinner or changes its resistance. The friction between rocks changes depending on how fast they are sliding past each other. This is called "rate dependence."

The Experiment: Smashing the Wall

The researchers built a massive computer simulation (a virtual earthquake lab) to test what happens when you account for this "honey-like" friction.

1. The Slow Test: When the crack moved slowly, the new theory matched the old one perfectly. Everything was normal.
2. The Speed Test: As they pushed the crack to move faster, approaching the edge of the "Forbidden Zone," the old rules started to crumble.
3. The Breakthrough: Instead of hitting a wall and jumping, the crack simply drove right through the Forbidden Zone.

The Analogy: The Car and the Speed Bump

Think of the old theory as a car hitting a massive speed bump (the Forbidden Zone). The theory said the car would have to stop, lift off the ground, and land on the other side.

The new theory shows that if the car has a special suspension (the changing friction), it doesn't need to jump. It can just drive smoothly over the bump, accelerating continuously from the slow lane, through the forbidden zone, and into the fast lane without ever stopping or jumping.

What This Means for Earthquakes

The paper concludes that:

• The "Forbidden Zone" isn't forbidden: Earthquakes can travel through the speed range between the surface wave and the shear wave smoothly and continuously.
• No sudden jumps needed: The dramatic "jump" from slow to super-fast earthquakes isn't always necessary. They can just speed up naturally as the friction changes.
• The Old Theory was too simple: It failed because it ignored the fact that friction changes when things move fast.

In short, the paper shows that the "speed limit" signs on the earthquake highway were wrong. Earthquakes can cruise through the middle speeds without needing a magical jump to get to the top speeds. This helps scientists understand how some massive earthquakes generate such intense shaking far away from the fault line.

Technical Summary

Technical Summary: Breakdown of the Classical Rupture Theory and Earthquake Propagation in the "Forbidden" Super-Rayleigh Range

Problem Statement
The propagation speed of earthquake ruptures is a critical determinant of seismic energy radiation and ground motion hazard. While it is established that some large earthquakes propagate as "super-shear" ruptures (exceeding the shear wave-speed, $c_s$), the transition mechanism from sub-shear to super-shear speeds remains a fundamental challenge in earthquake physics.

Classical two-dimensional (2D) rupture theory predicts a "forbidden" propagation range between the Rayleigh wave-speed ($c_R$) and the shear wave-speed ($c_s$). According to this theory, a rupture cannot propagate continuously through this super-Rayleigh range; instead, it must undergo a discontinuous "super-shear transition," often modeled via the nucleation of a secondary ("daughter") rupture ahead of the main ("mother") rupture. This prediction relies on specific assumptions regarding the elastodynamic fields and the frictional constitutive law, particularly the assumption that the residual frictional strength behind the rupture edge is independent of the slip rate.

Methodology
The authors revisit the assumptions underlying the classical 2D rupture theory to develop a generalized "Unconventional Singularities Theory" (UST). The methodology involves:

1. Theoretical Development:

   • The authors relax the classical assumption of rate-independent residual friction. Instead, they incorporate a linearized dependence of frictional strength ($\tau$) on slip rate ($v$), expressed as $\tau(v) \simeq \tau_0 + \eta_{\text{eff}} v$.
   • They derive generalized singular fields for the rupture edge, allowing for a non-universal singularity order $\xi$ (where $\xi \neq -1/2$). In classical theory, $\xi = -1/2$ is fixed by the rate-independent assumption.
   • They establish a relationship between the singularity order $\xi$, the rupture speed $c_r$, and the effective viscous-friction coefficient $\eta_{\text{eff}}$ via the boundary condition $\sigma_{xy} = \tau(v)$. This leads to a selection equation for $\xi$ that depends on the Rayleigh function $R(c_r)$.
   • They derive a generalized energy balance equation relating the energy release rate $G$ to the breakdown energy $G_f(\delta)$, which now scales with slip $\delta$ according to a power law determined by $\xi$.

2. Numerical Simulations:

   • Large-scale 2D boundary integral method simulations were performed using a nonlinear, rate-and-state friction law. This law features an N-shaped steady-state friction curve, consistent with experimental observations, where the high-slip-rate branch exhibits rate-strengthening behavior ($\eta_{\text{eff}} > 0$).
   • Ruptures were nucleated under various background stress levels ($\tau_d$) to control propagation speeds ranging from low sub-Rayleigh speeds ($c_r \approx 0.27 c_R$) to speeds exceeding $c_s$.
   • The simulations tracked spatiotemporal fields of slip rate, fault strength, and rupture speed to quantitatively test the theoretical predictions.

Key Contributions and Results

• Validation of UST in Sub-Rayleigh Regimes: The study demonstrates that the UST, which allows for $\xi \neq -1/2$, is in excellent quantitative agreement with numerical simulations for rupture speeds well below and close to the Rayleigh wave-speed ($c_R$). Specifically, the singularity order $\xi$ (measured to be approximately $-0.36$ to $-0.34$) and the breakdown energy scaling are self-consistently obtained from both the energy balance and the frictional boundary condition, independent of the classical $\xi = -1/2$ prediction.
• Breakdown of Classical Theory: The results confirm that the classical 2D rupture theory fails to quantitatively account for numerical data even at moderate speeds because it incorrectly assumes rate-independent friction.
• The "Forbidden" Range is Not Forbidden: As the rupture speed approaches $c_R$, the linear approximation of the friction law breaks down due to increasing nonlinearity. Consequently, the UST itself signals its own breakdown as $\xi \to 0$. Crucially, the numerical simulations reveal that the rupture does not undergo a discontinuous jump. Instead, it propagates continuously and smoothly through the "forbidden" super-Rayleigh range ($c_R < c_r < c_s$) and into the super-shear regime ($c_r > c_s$).
• Absence of Sharp Transition: The simulations show no evidence of a sharp super-shear transition (such as the "mother-daughter" mechanism) for the friction laws employed. The rupture speed increases smoothly, crossing $c_R$ and $c_s$ without divergence in slip rate or sign changes in stress fields that would be predicted by the classical theory with $\xi = -1/2$.

Significance and Claims
The paper claims that the existence of a "forbidden" super-Rayleigh speed range is not a universal feature of frictional rupture but is an artifact of the classical theory's assumption of rate-independent residual friction. By incorporating the generic, experimentally observed rate dependence of friction, the study demonstrates that:

1. Frictional ruptures can propagate continuously through the super-Rayleigh range.
2. The transition to super-shear speeds can occur smoothly without a discontinuous jump or the nucleation of secondary ruptures.
3. The classical 2D rupture theory has limitations when applied to fast earthquake propagation where frictional rate dependence is significant.

The authors note that while their results challenge the universality of the "forbidden" range, the specific nature of the super-shear emergence may depend on the detailed properties of the interfacial constitutive law. They conclude that frictional rate dependence has profound implications for understanding fast earthquake propagation and that future work should apply these findings to geophysical field observations and extend the theory to three dimensions.