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REVIEW 3 major objections 5 minor 74 references

Learning the nature of viscoelasticity in geologic materials with MCMC

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A single nonlinear Burgers-type viscoelastic model, calibrated by Bayesian inference from laboratory experiments, reproduces both steady and transient deformation of ice and olivine and predicts attenuation data it was not trained on.

desk verdict Good idea, plausible fits, but the attenuation prediction rests on a single fitted relaxation time and needs a baseline comparison before the mechanistic story holds. read the letter →

arxiv 2504.14028 v1 pith:L6JCXCOG submitted 2025-04-18 physics.geo-ph physics.comp-phphysics.data-an

Ron Maor , Lars Hansen , Douglas Jerolmack , David Goldsby This is my paper · ORCID
classification physics.geo-phphysics.comp-phphysics.data-an
keywords viscoelasticityBurgersmodelMarkovchainMonteCarlomicrostructureevolutionicerheologyolivineseismicattenuationBayesianinference
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

The paper attempts to show that one mechanistic model can span the full viscoelastic behavior of geologic materials: a nonlinear Burgers-type circuit whose parameters are learned from laboratory data by Markov chain Monte Carlo. The authors fit this model to constant-strain-rate compression tests on polycrystalline ice and to a forced-oscillation experiment on olivine, then use the ice calibration to predict the attenuation spectrum of a different ice experiment that was not part of the fitting data. If the claim holds, geodynamic and glaciological modelers would gain a single constitutive description that ties evolving grain size and crystal orientation to both steady flow and transient creep, and they could use lab-tuned models to forecast behavior in conditions experiments have not yet reached.

What carries the argument

The load-bearing object is the nonlinear Burgers model, a mechanical circuit in which a Maxwell element (an elastic spring in series with a nonlinear viscous dashpot) is connected in series with a Kelvin-Voigt element (a spring parallel to a dashpot). The Maxwell dashpot follows a dislocation-creep flow law with a stress exponent, an activation enthalpy, and explicit evolution of average grain size $d$ and crystal-orientation factor $F$; the Kelvin-Voigt element supplies the transient and anelastic response, linearized from a hyperbolic-sine backstress law on the assumption that the effective stress on the transient dashpot is small. This one circuit carries the entire argument because every deformation mode — elastic, transient, and steady-state flow — is represented in the same equations, so a single posterior over the parameters must simultaneously satisfy loading, oscillation, and attenuation observations.

What would settle it

Run forced-oscillation experiments on ice at elevated median stresses around 1.5, 2, and 4 MPa, where the paper predicts increasing attenuation and a flattening of the attenuation peak; if the peak persists instead of being suppressed at high stress, or if energy loss does not depend on oscillation amplitude, the single-element transient branch is wrong. A sharper test is the width of the attenuation peak at one stress: a single relaxation time gives a peak of one fixed width, while a distribution of relaxation times gives a broader peak.

Watch

Extended reading notes

Core claim

The central claim is that a nonlinear Burgers model with a power-law Maxwell dashpot and a single linear Kelvin-Voigt element, augmented with phenomenological relaxation equations for grain size and crystallographic preferred orientation, is sufficiently identifiable from laboratory data to represent ice and olivine in the dislocation-creep regime. After Bayesian inference, the posterior distributions for the stress exponent and the critical strain for grain-size evolution converge to physically plausible values ($n \approx 4$ and $\varepsilon_{c1} \approx 0.03$ for ice; $n \approx 3$ for olivine), the model captures the peak stress and steady-state flow of ice stress-strain curves and the amplitude and phase of olivine forced oscillations, and the calibrated ice model reproduces the shape of an attenuation spectrum measured in a separate experiment, including a Debye peak and a power-law high-temperature background. The paper further claims predictive power: extrapolating the ice model to higher median stresses yields amplitude-dependent attenuation with an increasing power-law background and a suppressed Debye peak, which future experiments could verify.

Load-bearing premise

The load-bearing premise is that one spring-and-dashpot element, with a single stiffness and a single viscosity, is enough to represent the short-term and recoverable deformation of both ice and olivine in the dislocation-creep regime, so that any spread of relaxation times comes from the nonlinear flow law and evolving grain structure rather than from a distribution built into the model.

Editorial extensions

If this is right

  • Geodynamic and ice-sheet models could replace separate steady-state and transient parametrizations with a single nonlinear Burgers law whose transient branch is tied to measurable microstructure.
  • The calibrated ice model can generate attenuation spectra for stress amplitudes and frequencies not yet explored in the laboratory, directly informing tidal-dissipation models for icy moons and ice-shelf flexure.
  • Posterior convergence for individual parameters, such as the Kelvin-Voigt spring and dashpot, reveals which deformation mechanisms a given experiment actually constrains, guiding the design of future transient-creep and attenuation experiments.
  • The same framework can be applied to other deformation mechanisms and materials by changing the stress and grain-size exponents, allowing composite flow laws to be tested against several experimental geometries at once.

Reading between the lines

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

  • The claim that nonlinearity and microstructure evolution broaden the relaxation spectrum beyond the single spring-and-dashpot element is asserted rather than demonstrated; computing the effective relaxation-time distribution from the model's small-oscillation response at different mean stresses would test it directly.
  • Because the predicted attenuation peak is the frequency-domain image of a single relaxation time, a high-resolution attenuation measurement that resolves the peak width would discriminate this model from models with a continuous relaxation spectrum.
  • The same inference pipeline could be pointed at other strain-rate histories, such as stress-relaxation or stress-drop experiments, to check whether the posterior parameters learned from constant strain rate and forced oscillation agree out of sample.
  • The same inference approach could be used to test uniqueness: fitting experiments with different transient loading histories and checking whether the inferred spring-and-dashpot values coincide would show how much of the model's structure is actually demanded by the data.
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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 / 5 minor

Summary. The authors combine a nonlinear Burgers-type viscoelastic model with Hamiltonian Monte Carlo (No-U-Turn Sampler) inference, incorporating evolving grain size and crystallographic preferred orientation (CPO) in a Maxwell dashpot with a power-law flow law and a single linear Kelvin-Voigt element for transient/anelastic response. They fit three constant-strain-rate stress-strain experiments on polycrystalline ice and a forced-oscillation experiment on olivine, reporting posterior distributions and retrodiction bands. They then use the ice-calibrated model to compute an attenuation spectrum for comparison to McCarthy et al. (2016), finding a qualitative Debye-peak shape and nonlinear amplitude dependence but quantitative deviations from the measured attenuation. The paper claims that the method constrains a nonlinear viscoelastic model that captures both steady and transient dynamics and can predict dynamics for untrained data.

Significance. The paper is valuable in demonstrating the feasibility of Bayesian calibration of a microstructurally informed Burgers model against two distinct laboratory datasets, and the ice retrodiction bands and olivine amplitude/phase fits are encouraging. The model also makes falsifiable predictions for attenuation at elevated stress levels. If the predictive claim were quantitatively established, the approach could improve geodynamic modeling by providing parameter uncertainties and a systematic route from laboratory creep and oscillation data to constitutive laws. However, the central predictive claim is not yet established: the out-of-sample attenuation comparison is qualitative, and the use of a single Kelvin-Voigt relaxation time means that the predicted spectral shape is largely fixed by fitted parameters rather than independently tested. The paper's own caveats and the absence of comparison to simpler baselines leave the mechanistic interpretation underdetermined. With additional model-comparison and posterior-predictive diagnostics, the framework would be a solid contribution.

major comments (3)
  1. [Predictions, Fig. 6] The central claim that the model "can also predict dynamics for data it was not trained on" is not quantitatively supported by the out-of-sample attenuation test. The model curves in Fig. 6 visibly deviate from the McCarthy et al. (2016) data, and the manuscript's explanations (GBS regime, ambient pressure, cracking, low-frequency microstructure evolution) are plausible but unverified. With only a handful of data points and no uncertainty bands on the predicted attenuation, the comparison establishes at most a qualitative resemblance in the form of a Debye peak and high-temperature background. The authors should either quantify the prediction error and propagate posterior uncertainty into the attenuation spectra, or soften the abstract's predictive claim to reflect that the extrapolation is qualitative.
  2. [Transient Component and Discussion] The transient/anelastic element is a single linear Kelvin-Voigt circuit, so the model has exactly one relaxation time tau_K = eta_K/E_K. The position and shape of the Debye peak in Fig. 6 are therefore largely determined by the fitted values of eta_K and E_K; the assertion that the nonlinear dashpot and microstructure evolution broaden the relaxation spectrum is not demonstrated. A simpler linear Burgers model (or a two-Kelvin-element version) fitted with the same MCMC approach might reproduce the ice and olivine data equally well while yielding a different attenuation spectrum. Reporting posterior-predictive checks and comparing information criteria against such baselines is necessary before the mechanism can be attributed to the nonlinear/microstructural ingredients.
  3. [MCMC with Ice Data and Materials and Methods] The model has 11 free parameters, but the ice calibration uses only three stress-strain curves, and the paper itself notes (Discussion) that a good fit has high probability if the prior space is too large. The posterior convergence of E_K and eta_K is reported, but no posterior correlations or identifiability analysis is shown. The statement that the model captures the experiment "uniquely with this mechanistic formulation" (Results, ice) is stronger than the evidence. The authors should report posterior correlation matrices and, at minimum, a complexity measure such as the effective number of parameters to show that the data can constrain the 11-dimensional parameter vector.
minor comments (5)
  1. [Predictions] The text says the attenuation data are "sometimes reffers to as high-temperature background"; "reffers" should be "referred" and the phrasing should be corrected.
  2. [Discussion & Conclusion] "Despite these incogruencies" should be "incongruencies" or "inconsistencies".
  3. [Figure 3 caption] The caption states that the temperature in the numerical solutions was 260 K, while the main text reports experiments at 263 K; please clarify whether this difference is intentional.
  4. [MCMC with Olivine Data] The olivine strain data were detrended by fitting and subtracting a polynomial before computing the oscillatory strain rate; the uncertainty of this correction is not propagated into the MCMC, and its influence on the inferred Kelvin-Voigt parameters should be discussed.
  5. [Olivine Data and Priors] The text says grain-size evolution was incorporated and later states that microstructural analysis showed negligible grain-size evolution; the non-convergence of the grain-size parameter in Fig. S5B is consistent with that, but the main text should state this more directly to avoid apparent contradiction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: MCMC fits are clearly separated from the external attenuation benchmark, and the single-Kelvin-element Debye peak is an acknowledged model feature rather than an inferred result.

full rationale

The paper's derivation chain is not circular. The model parameters, including the Kelvin-Voigt EK and etaK, are obtained by MCMC fits to two distinct experimental datasets: constant-strain-rate ice experiments (Qi & Goldsby) and forced-oscillation olivine experiments. The ice attenuation spectrum in Fig. 6 is then computed with the optimized ice model and compared to McCarthy et al. (61), which was not used in the MCMC; the paper explicitly lists several reasons for deviations from those data, confirming that the comparison is an external, out-of-sample benchmark rather than a fitted target. The Debye peak is indeed a mathematical consequence of the single Kelvin-Voigt element (tauK = etaK/EK), but the paper does not present that peak as evidence that the Kelvin element was learned from the attenuation data; rather, it acknowledges the single-element assumption and discusses Andrade-type alternatives as a limitation (Discussion & Conclusion: 'Other viscoelastic models, such as the Andrade model, include an infinite series of dashpots and springs in the anelastic circuit to broaden the relaxation spectrum and better fit laboratory data'). The microstructure-evolution equations and the sinh backstress law are taken from prior published work (some co-authored by the present authors), but those are stated modeling assumptions or data sources with independent experimental/theoretical content, not uniqueness theorems or unverified self-citations invoked to forbid alternatives. The paper's own Discussion warns that 'reaching a good fit has a high probability if the parameter space of the priors is too large,' which is a model-complexity and overfitting caveat, not a circular step. No equation is shown to reduce to its own input, no fitted parameter is renamed as an independent prediction, and the external attenuation benchmark is genuinely held out from the fitting procedure.

Assumptions & free parameters 11 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several modeling choices imported from prior literature: the power-law flow law with CPO softening, the single linear Kelvin-Voigt transient element, and the phenomenological microstructure evolution equations. These are not derived from first principles in this paper; the MCMC only calibrates their parameters. The Bayesian framework contributes uncertainty quantification but cannot validate the model form itself.

free parameters (11)
  • etaK (Kelvin-Voigt dashpot viscosity) = Posterior; prior U[1e9, 5e11] Pa·s (ice), U[5e8, 1e12] Pa·s (olivine)
    Sets the anelastic relaxation time; no direct lab values in dislocation creep, so sampled widely.
  • EK (Kelvin-Voigt spring modulus) = Posterior; prior U[5e8, 1e10] Pa (ice), U[7e8, 9e10] Pa (olivine)
    Kelvin spring stiffness; converged to a narrow normal posterior for olivine (Fig S5C).
  • epsilon_c1 (critical strain for grain-size evolution) = Ice posterior approximately 0.03; olivine non-convergent (Fig S5B)
    Controls how quickly grain size relaxes toward steady-state; motivated by recrystallization literature.
  • epsilon_c2 (critical strain for CPO development) = Ice posterior; prior U[0.01, 0.5]
    Controls CPO evolution rate; used only for ice, since olivine showed no CPO change.
  • dss (steady-state grain size) = Posterior; prior U[10, 200] µm (ice)
    Target grain size in evolution law; prior from Qi and Goldsby (2021).
  • Fss (steady-state CPO factor) = Posterior; prior U[0.1, 0.9] (ice)
    Steady-state geometric softening factor; wide prior because no previous record.
  • n (stress exponent) = Ice approximately 4; olivine approximately 3
    Stress exponent in flow law; converged within literature priors.
  • p (grain-size exponent) = Ice posterior within U[0.1, 0.7]; olivine U[0.2, 1.5]
    Grain-size exponent in flow law; sampled with literature priors.
  • QM (activation enthalpy) = Ice approximately 65 kJ/mol (narrow prior); olivine approximately 445 kJ/mol
    Activation energy for creep; constrained by literature priors.
  • beta (flow-law material parameter) = Ice approximately 7.29e-4 J-1 (narrow prior); olivine approximately -8.6e-5 J-1
    Combines pre-exponential factor and QM; narrowed to allow convergence.
  • EM (Maxwell elastic modulus) = Ice prior N(4, 2) GPa; olivine N(200, 10) GPa
    Instantaneous elastic modulus; sampled from literature priors.
assumptions (5)
  • domain assumption The Maxwell dashpot follows the power-law flow law with geometric softening (Eq. 1): epsilon_dot_Md = (sigma/F)^n d^(-p) exp(QM(beta - 1/RT)).
    Standard dislocation creep law; the multiplicative 1/F^n softening dependence on CPO is taken from prior work, not derived here.
  • domain assumption A single linear Kelvin-Voigt element represents the transient and anelastic response, justified by linearizing the sinh backstress law (Eq. 2).
    The paper acknowledges that Andrade-type models with a spectrum of relaxation times fit laboratory data better; the single KV element is a simplification.
  • domain assumption Grain size and CPO evolve according to phenomenological relaxation equations (Eqs. 3-4) from Hansen et al. (2012).
    These are heuristic relaxation laws with characteristic strains epsilon_c1 and epsilon_c2 treated as free parameters.
  • standard math Observational model D = M(x, theta) + epsilon with additive error; Bayesian likelihood follows from this model.
    Standard statistical assumption; the error distribution and its variance are not explicitly reported in the main text.
  • standard math NUTS/HMC samples from the posterior with 5 chains of 3,000 accepted samples each; chain convergence is assumed.
    Standard MCMC practice; no explicit convergence diagnostics such as R-hat are reported in the main text.

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Pith. "Pith review of Learning the nature of viscoelasticity in geologic materials with MCMC." pith.science (2026). https://pith.science/paper/L6JCXCOG

@misc{pith2026250414028,
  author       = {Pith},
  title        = {Pith review of: Learning the nature of viscoelasticity in geologic materials with MCMC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6JCXCOG}},
  note         = {Machine review of arXiv:2504.14028}
}
read the original abstract

Rock and ice are ubiquitous geologic materials. While apparently solid, they also exhibit fluid behavior under stress - a property termed viscoelasticity. Viscoelastic convection of Earth's mantle drives tectonic plate motion with consequences for earthquakes and sea-level rise, while viscoelastic deformation of ice controls glacier flow and the flexure of icy moons. For crystalline materials, "flow laws" describing bulk rheology can be derived from understanding microstructural dynamics such as crystal-defect migration. Common geologic materials like ice and olivine have grain sizes and crystal orientations that evolve with strain; this complexity precludes a first principles approach. Here we use a Bayesian inference method to learn the connection between microstructure and flow in ice and olivine, from fits to experimental data of these materials undergoing steady-state deformation and forced oscillations. We demonstrate that this method can constrain a nonlinear viscoelastic model for each material, that is capable of capturing both steady and transient dynamics and can also predict dynamics for data it was not trained on. Our results may improve geodynamic models that rely on parameterized constitutive equations, while our approach will be useful for experimental design and hypothesis testing.

Figures

Figures reproduced from arXiv: 2504.14028 by the authors.

Figure 1
Figure 1. The role of viscoelasticity in geodynamic phenomena. (A) The tidal response of the icy shell of Jupiter’s moon Europa is modeled using viscoelastic models (e.g. 43, 44). Picture of Europa from NASA Image and Video Library (45) (B) Viscoelastic properties are incorporated in studies of outlet glacier flows and ice shelf flexure (e.g. 46, 47). Picture of a small valley glacier exiting the Devon Island Ice Cap, from NA… view at source ↗
Figure 2
Figure 2. The nonlinear Burgers model of this study. It has the same circuit as the linear Burgers model, except that the Maxwell dashpot represents a nonlinear flow law with evolving microstructure. substantial uncertainty remains surrounding the microme￾chanical processes associated with transient creep, mainly due to experimental constraints. The limited duration of the transient phase, the constantly changing microstructu… view at source ↗
Figure 3
Figure 3. (A) The constant strain rate ice experiment: the sample is deformed using a stepper motor that pushes the sample from the bottom at a constant rate. (B-D) Stress-strain curves from MCMC runs, along with data from (58). The experiments are referred to as PIL40, PIL72, and PIL105 in the original text. The red curve is the original experimental data set. The blue dashed curve is the nonlinear Burgers model solution wit… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Priors and posterior distributions of the MCMC runs of the PIL40 ice experiment of (58) (Figure 3B). Prior distributions are represented by red lines, according to [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Experimental setup and MCMC results for the olivine data. (A) The olivine sample was deformed using axial forced oscillations in the Deformation￾DIA apparatus. (B) Stress data accompanied by the corresponding numerical results. The red line denotes the experimental dat…
Figure 6
Figure 6. Figure 6: Attenuation spectrum from the MCMC model for ice, for several median stress amplitudes on a log-log scale. Blue curve is for σm = 1 MPa and σ0 = 0.17 MPa, corresponding to the experiments of McCarthy et al. (61). The red curve is σm = 1.5 MPa and σ0 = 0.5 MPa; green cu…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.