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Determination of Effect of the Movement of a Finite, Dip-slip Fault in Viscoelastic Half-space of Fractional Burger Rheology

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Pith's one-line read The paper derives closed-form displacement, stress, and strain for a finite buried dip-slip fault creeping in a fractional Burger viscoelastic half-space, finding inclination and creep velocity dominate the response.

desk verdict Finite-fault fractional Burger solution, but post-fault Green's function is reflected across the wrong plane, so stresses don't vanish on the free surface. read the letter →

arxiv 2506.03977 v1 pith:7ID3BQ3N submitted 2025-06-04 physics.geo-ph

Pabita Mahato , Seema Sarkar (Mondal) This is my paper
classification physics.geo-ph MSC 26A3374D1086A1586A17 PACS 91.30.Dk91.30.-f
keywords fractionalBurgerrheologydip-slipfaultviscoelastichalf-spaceCaputoderivativeMittag-LefflerfunctionGreen'saseismicdeformationstressaccumulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper works out, in closed form, the displacement, stress, and strain produced in the aseismic interval around a finite, buried, inclined dip-slip fault when the surrounding half-space obeys fractional Burger rheology. Such slow creep deformation is what geodetic instruments see between earthquakes, so explicit formulas for it are a step toward reading stress accumulation from surface data. The derivation combines fractional Laplace transforms, a modified Green's function for a half-space, and the correspondence principle, with results expressed through Mittag-Leffler functions. The paper's main finding is that fault inclination and creep velocity change the response strongly, while the order of the fractional derivative has a moderate effect.

What carries the argument

The load-bearing object is the fractional Burger constitutive equation $$\tau_{ij}+p_1\,{}_0D_t^\$\alpha$\tau_{ij}+p_2\,{}$_0D_t^{{2\alpha}}$\tau_{ij}=2q_1\,{}_0D_t^\$\alpha$ e_{ij}+2q_2\,{}$_0D_t^{{2\alpha}}$e_{ij},$$ with $p_1,p_2,q_1,q_2$ built from two viscosities and two rigidities and ${}_0D_t^\alpha$ the Caputo fractional derivative. The argument runs by Laplace transforming this equation, solving the resulting harmonic equations $\nabla^2 U_i=0$ with boundary conditions in the transformed domain, applying the correspondence principle, and inverting with the help of the Mittag-Leffler function $E_\alpha$. After fault movement begins, a modified Green's function of Maruyama type, with an image term to satisfy the free-surface condition, converts the prescribed vertical slip $V t_1 f$ into the displacement integral $\Psi_1$, and differentiation of that integral supplies $\Psi_2$, $\Psi_3$, and $\Psi_4$ entering the shear stresses.

What would settle it

Compare the model's post-seismic stress field with the exact elastic dislocation solution for a vertical dip-slip fault in a homogeneous half-space with a nonzero Poisson ratio: if that exact solution gives nonzero horizontal surface displacements and nonzero $\tau_{11}$, $\tau_{12}$, and $\tau_{22}$, then the simplification $(u_1)_2=(u_2)_2=0$ collapses and the stress magnitudes in Eq. (49) would change.

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Extended reading notes

Core claim

The central claim is that for a fault of finite length $2L$ and width $D$, buried at depth $d$ and inclined at angle $\theta$, the full set of displacement, stress, and strain components after creep begins can be written analytically as Eqs. (48)-(50): the background aseismic solution plus a fault-movement part in which vertical displacement is $V t_1 H(t_1)\Psi_1$, vertical shear stresses are proportional to Mittag-Leffler relaxation functions, and horizontal displacement components are set to zero. The solution exhibits stress accumulation before failure, with the fault starting to move when $\tau_{22}$ reaches a critical value (taken as 150 bar; about 160.5 years for $\theta=\pi/6$, $\alpha=0.5$), followed by slow stress relaxation. The authors conclude that changing creep velocity and fault inclination produces significant differences in displacement, stress and strain, whereas changing fractional order $\alpha$ produces only moderate differences.

Load-bearing premise

After fault movement begins, the model assumes the displacement has only a vertical component because the fault is dip-slip, and the derived zero horizontal stresses follow from that assumption.

Editorial extensions

If this is right

  • At the chosen parameters, peak surface displacement is about $2.1\times10^{-6}$ m/yr for $\theta=\pi/6$ and $1.6\times10^{-6}$ m/yr for $\theta=\pi/2$, so shallower dips give larger near-fault uplift.
  • Surface shear stress $\tau_{23}$ accumulates to order $10^6$ N/m$^2$ after creep begins, peaking roughly 2.5 km from the fault and scaling with creep velocity.
  • The pre-fault stress component $\tau_{22}$ reaches the critical threshold at about 160.5 years for $\theta=\pi/6$, $\alpha=0.5$, setting the model's time of fault movement.
  • Strain $e_{33}$ peaks near 11.1 km depth with magnitude about $2.3\times10^{-8}$, and the pattern matches observed strain-rate magnitudes.
  • Changing the fractional order $\alpha$ from 0.1 to 1 alters the stress-relaxation rate, with the most visible decrease in $\tau_{33}$ occurring for larger $\alpha$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: dropping the horizontal post-fault displacements $(u_1)_2=(u_2)_2=0$ removes Poisson-coupled deformation, so the zero values of $\tau_{11}$, $\tau_{12}$, and $\tau_{22}$ in Eq. (49) are best read as a modeling choice; an elastic solution with standard Poisson coupling would tell how large the omitted terms are.
  • Editorial inference: the fractional order $\alpha$ acts as a memory parameter with a mild effect in these plots, so the model offers a way to estimate $\alpha$ from geodetic time series of shear-stress relaxation, though no inversion is performed here.
  • Editorial inference: because the Green's-function representation is linear, the same machinery extends to interacting or non-planar faults by changing the slip function and integration domain, a direction the authors flag as future work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a quasi-static mathematical model for a finite, buried, inclined dip-slip fault in a viscoelastic half-space with fractional Burger rheology. The deformation is split into a pre-seismic phase, solved by Laplace-transformed trial displacements, and a post-seismic phase, solved by a Green's function technique and the correspondence principle. The authors present closed-form expressions for displacement, stress, and strain in Eqs. (48)-(50) and numerically study the effects of fault inclination, creep velocity, and fractional-derivative order. The central claim is that these analytical solutions describe how stress and strain accumulate and release during the aseismic period.

Significance. If the analytical solutions were valid, the paper would extend classical dislocation-based geophysical models to fractional Burger rheology and would provide a useful parametric tool for interpreting aseismic deformation. The paper also contains a concrete numerical prediction (the 160.5-year time to critical stress) and a systematic parametric study. However, the central claim is not established because the post-seismic Green's function is constructed for the wrong image plane and because the assumed displacement field is inconsistent with the equilibrium equations and boundary conditions. These are load-bearing issues: they affect the derivation of Eqs. (48)-(50), not merely the presentation.

major comments (3)
  1. [Section 3.2, Eq. (39)] The Green's function G1 in Eq. (39) places the image term at (y1 + ξ1, y2 − ξ2, y3 − ξ3), i.e., reflection across the plane y1 = 0. The model domain is y3 ≥ 0 with the free surface at y3 = 0, so the image source for a half-space problem must be placed at (y1 − ξ1, y2 − ξ2, y3 + ξ3) or otherwise constructed to make the tractions vanish on y3 = 0. As a result, the shear stresses (τ13)_2 and (τ23)_2 derived from this Green's function do not vanish at y3 = 0, in direct violation of boundary condition (8). Figures 4-6 confirm this by plotting nonzero "surface" shear stress τ23 at y3 = 0. Since the post-fault displacement in Eq. (45) and the stress formulas in Eq. (46) all inherit this Green's function, the final solution (48)-(50) does not solve the boundary value problem stated in Section 2.3.
  2. [Section 3.2, Eq. (37)] The assumption (u1)_2 = (u2)_2 = 0 is not a consequence of dip-slip kinematics; a vertical dislocation in a homogeneous elastic or viscoelastic half-space produces horizontal displacements through Poisson coupling. More seriously, with (τ11)_2 = (τ12)_2 = (τ22)_2 = 0, the equilibrium equations (3) reduce to ∂(τ13)_2/∂y3 = 0 and ∂(τ23)_2/∂y3 = 0. The expressions for (τ13)_2 and (τ23)_2 in Eq. (46) depend on y3 through Ψ2 and Ψ3, so they violate the equilibrium equations. This is an internal inconsistency in the post-fault solution, independent of the Green's-function issue.
  3. [Section 3.2, Eqs. (41)-(42)] The coordinate transformation in Eq. (41) is not the inverse of Eq. (1): from Eq. (1), ξ3 = −ξ′2 cosθ + ξ′3 sinθ + d, whereas Eq. (41) has +ξ′2 cosθ. On the fault surface, where ξ′2 = 0, the error does not change the final integrand, but the stated transformation is incorrect. In addition, the derivation of the sinθ factor in Eq. (42) is not documented; it should follow from a Jacobian computation for dξ3 dξ1 in terms of dξ′3 dξ′1. As written, the passage from Eqs. (38)-(40) to Eq. (42) is not reproducible, and this step defines Ψ1 and hence the post-fault displacement.
minor comments (4)
  1. [Section 3.2, Eqs. (36) and (38)] The dislocation condition is stated for [u3]_F in Eq. (32), but Eqs. (36) and (38) use [u1]_F in the Laplace transform and in the Green's function integrand; this appears to be a typo that should be corrected.
  2. [Section 4, slip function] The slip function is written as f(y′1, y′3) = R(1 − 1/(L^2 y′1^2))(1 − 3y′3^2/D^2 + 3y′3^3/D^2); as printed this factor is dimensionally inconsistent and singular at y′1 = 0. Presumably a factor such as (1 − y′1^2/L^2) was intended.
  3. [Table 1] The table lists T = 160.5 years as an input parameter, but Section 3.1 states that this value is derived from the expression for (τ22)_1; the table should distinguish input parameters from derived results.
  4. [Figures 4-6] The captions refer to "surface shear stress", but boundary condition (8) requires τ23 = 0 on the free surface y3 = 0; the nonzero plotted values therefore either reveal the boundary-condition violation or the plots are made at some other depth, which should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: model parameters are external inputs and the 160.5-year critical time is computed from the derived stress expression, not fitted.

full rationale

The analytical derivation is self-contained in the sense that the central equations (48)-(50) are obtained by solving the stated constitutive equations and boundary conditions, not by fitting parameters to the model's outputs. The rheological parameters (rigidities, viscosities, initial stresses, fault geometry, creep velocity) are taken from published field-based sources (Cathles 1975; Clift 2002; Aki 1980; Karato 2010; Kundu et al. 2021). The 160.5-year time to critical stress is computed by solving the derived expression for (τ22)1 against the input threshold τc=150 bar, so it is a model output rather than an input used to construct the solution. The post-fault displacement is obtained from the standard Maruyama Green's function technique, an independent external method, and the assumed slip function is explicitly introduced as a modeling input, not as a derived consequence. Self-citations (Mahato et al. 2022; Mahato and Sarkar Mondal 2025) appear only as background/precedent, and the slip function from Kundu et al. (2021) is an assumption rather than a load-bearing theorem. No equation reduces by construction to its own inputs. The apparent image-reflection inconsistency in Eq. (39) (image across y1=0 rather than the free surface y3=0) and the resulting failure of boundary condition (8) are internal correctness/boundary-condition concerns, not instances of circular reasoning.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The derivation relies on the fractional Burger constitutive law (Eq. 2), the quasi-static equilibrium assumption (Sec. 2.2), a linear spatial ansatz for pre-seismic displacement (Eq. 17), and Maruyama's half-space Green's function. Two ad hoc constants, k and R, enter the remote-stress boundary condition and slip amplitude. No new physical entities are postulated.

free parameters (2)
  • k = 10^-9
    Constant in the remote stress boundary conditions (Eqs. 6-7) that makes τ∞ grow slowly with time; chosen as a very small value without justification from data.
  • R (magnifying factor) = 1 cm
    Scaling factor in the slip function f(y'1,y'3) (Section 4), chosen to set units/amplitude of slip; not constrained by observations.
assumptions (6)
  • domain assumption The inertia terms in the stress equation of motion are negligible (quasi-static equilibrium).
    Section 2.2 states inertial forces are negligible and body forces do not change, reducing the momentum equation to equilibrium.
  • domain assumption No change in body forces occurs relative to the initial stress level.
    Section 2.2: 'no change of body forces is supposed to occur'.
  • domain assumption The fractional Burger constitutive law (Eq. 2) with Caputo fractional derivatives of order α describes the viscoelastic material.
    Assumed based on Okuka and Zorica (2020) and Segall (2010); no experimental validation for the specific fault setting.
  • ad hoc to paper Trial solutions for the pre-seismic displacement are linear in coordinates (Eq. 17).
    The displacement solutions are assumed to be A_i y1 + B_i y2 + C_i y3 plus initial terms; this restricts the solution to a spatially linear stress state and is not derived from the boundary value problem.
  • standard math Maruyama's Green's function for a point force in a half-space gives the displacement from a dislocation via Eq. (38).
    Uses the classical image-source Green's function (Eqs. 39-40); standard in dislocation theory, though the 'modified' aspect is not explained.
  • standard math The correspondence principle applies to fractional constitutive equations under Laplace transformation.
    Used throughout Section 3 to convert time-domain constitutive equations to algebraic equations in the Laplace domain; valid for linear viscoelasticity with Caputo derivatives under zero initial conditions.

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Cite this review

Pith. "Pith review of Determination of Effect of the Movement of a Finite, Dip-slip Fault in Viscoelastic Half-space of Fractional Burger Rheology." pith.science (2026). https://pith.science/paper/7ID3BQ3N

@misc{pith2026250603977,
  author       = {Pith},
  title        = {Pith review of: Determination of Effect of the Movement of a Finite, Dip-slip Fault in Viscoelastic Half-space of Fractional Burger Rheology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ID3BQ3N}},
  note         = {Machine review of arXiv:2506.03977}
}
read the original abstract

The seismically active regions often correlate with fault lines, and the movement of these faults plays a crucial role in defining how stress is stored or released in these areas. To investigate the deformation and accumulation/release of stress and strain in seismically active regions during the aseismic period, a mathematical model has been developed by considering a finite, creeping dip-slip fault inclined in the viscoelastic half-space of a fractional Burger rheology. Laplace transformation for fractional derivatives, Modified Green's function technique, correspondence principle and finally, the inverse Laplace transformation have been used to derive analytical solutions for displacement, stress and strain components. The graphical representations were depicted using MATLAB to understand the effect on displacement, stresses and strains due to changes in inclinations and creep velocities of the fault, as well as orders of the fractional derivative. Our investigation indicates that a change in creep velocity and inclination of the fault has a significant effect, while a change in the order of fractional derivative has a moderate effect on displacement, stress, and strain components. Analysis of these results can provide insights into subsurface deformation and its impact on fault movement, which can lead to earthquakes.

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