Deformation Due to a Pair of Infinite and a Pair of infinite-finite planar faults in Fractional Standard Linear Solid

Technical Summary: Deformation Due to a Pair of Infinite and a Pair of Infinite-Finite Planar Faults in Fractional Standard Linear Solid

Problem Statement
Seismically active regions frequently contain multiple interacting faults where slip along one fault significantly influences stress accumulation and distribution in neighboring faults. While previous studies have modeled fault interactions using integer-order derivatives within viscoelastic media (e.g., Maxwell or Standard Linear Solid models), these traditional approaches often overlook the "memory" effects inherent in geological materials. This paper addresses the deformation of the Earth's lithosphere-asthenosphere system during the aseismic period by modeling two specific scenarios involving interacting creeping strike-slip faults:

1. Case-I: The interaction between two infinite buried faults.
2. Case-II: The interaction between one infinite buried fault and one finite surface-breaking fault.

The study posits that the Earth's medium behaves as a viscoelastic half-space characterized by a Fractional Standard Linear Solid (FSLS) model, which better captures the time-dependent rheological behavior and memory effects of crustal rocks compared to integer-order models.

Methodology
The authors developed theoretical mathematical models to derive analytical solutions for displacement, stress, and strain components. The methodology involves the following key steps:

• Mathematical Formulation: The medium is defined using a fractional standard linear solid constitutive equation involving the Caputo fractional derivative operator (${}_{0}D^{\alpha}_{t}$) where the order $\alpha \in (0, 1]$. The stress-strain relations incorporate effective rigidity ($\mu$) and effective viscosity ($\eta$).
• Governing Equations: The analysis assumes quasi-static deformation (neglecting inertial forces). The stress equations of motion and boundary conditions are established for both infinite and finite fault geometries.
• Solution Technique: The authors employ the Laplace transformation to handle the time-dependent fractional differential equations. The solutions are derived using:
  • The Correspondence Principle (Rybicki, 1971).
  • A Modified Green's Function technique (Maruyama, 1966) to account for dislocation conditions across the fault planes.
  • The Mittag-Leffler function ($E_{\alpha}$), which naturally arises from the inverse Laplace transformation of the fractional operators.
• Scenarios Analyzed:
  • Solutions are derived for the state before fault creep (accumulation of tectonic stress) and after fault creep (dislocation due to slip).
  • Creep velocities ($V_1, V_2$) and depth-dependent creep functions are applied to simulate the movement of faults $F_1$ and $F_2$.
  • Numerical computations are performed using parameters derived from geological data (e.g., rigidity $\mu = 3.5 \times 10^{10}$ N/m², viscosity $\eta = 5 \times 10^{19}$ Pa·s) to generate graphical results for various fault inclinations ($\theta$), fractional orders ($\alpha$), and creep velocities.

Key Contributions

1. Fractional Modeling of Fault Interaction: The paper extends existing fault interaction models by incorporating fractional calculus, allowing for a more accurate representation of the memory-dependent viscoelastic behavior of the lithosphere.
2. Analytical Solutions for Complex Geometries: The study provides closed-form analytical expressions for displacement, stress, and strain fields for two distinct configurations: a pair of infinite buried faults and a mixed system of infinite and finite surface-breaking faults.
3. Quantification of Memory Effects: The research explicitly demonstrates how the fractional order $\alpha$ influences the accumulation and release of stress, linking the mathematical parameter to the physical "memory" of the geological medium.

Results and Numerical Findings
The numerical analysis, supported by graphical representations, yields several specific observations:

• Displacement Magnitudes: The change in surface displacement near the faults is of the order of $10^{-3}$ m/year. This aligns well with observed geophysical data (e.g., USGS reports of 2–6 mm/year), whereas previous integer-order models predicted values in the range of $10^{-5}$, showing a discrepancy with real-world observations.
• Influence of Fault Geometry:
  • In Case-I (two infinite faults), the displacement magnitude near the second fault is higher than near the first.
  • In Case-II (infinite and finite faults), the displacement magnitude near the finite fault is lower than near the infinite fault.
  • Displacement peaks near the fault trace and diminishes as distance increases, eventually tending to zero.
• Effect of Fractional Order ($\alpha$): The study finds that the rate of stress accumulation and release is highly dependent on $\alpha$. Lower values of $\alpha$ (indicating stronger memory effects) result in higher magnitudes of shear stress accumulation and slower relaxation. As $\alpha$ increases toward 1 (approaching integer behavior), the stress release rate increases.
• Driving Force and Stress Distribution: The total driving force across the faults shows rapid release after the onset of creep. The magnitude of this release varies with fault inclination and creep velocity. Contour maps reveal that stress accumulation is most significant near the free surface and fault dislocation zones, while stress reduction is prominent at depths corresponding to the fault locations.
• Strain Analysis: Shear strain ($e_{12}$) variations are found in the range of $10^{-4}$, consistent with data from real-world events like the 2011 Tohoku-oki earthquake.

Significance and Claims
The authors claim that their fractional model offers a more general and accurate representation of fault interaction dynamics compared to traditional integer-order models. Specifically:

• The fractional approach successfully captures the memory effects and nonlinearity of viscoelastic media, which are often overlooked in standard models.
• The model's predictions for displacement rates ($10^{-3}$) show strong agreement with observed geophysical data, suggesting that fractional calculus is a superior tool for predicting seismic behavior and subsurface deformation.
• The findings indicate that the interaction between faults influences the stress field more extensively than a single fault, and that the fractional order $\alpha$ is a critical parameter in determining the rate of stress release.

The paper concludes that these theoretical models enhance the understanding of subsurface deformation and its role in influencing fault dynamics and earthquake generation, providing a robust framework for analyzing aseismic periods in seismically active zones.