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REVIEW 4 major objections 5 minor 39 references

Strain dependent viscous response describes the mechanics of cohesionless soft granular materials

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that adding a strain-dependent prefactor to the viscous branch of a standard linear solid model reproduces cyclic loading of cohesionless soft granular packings, including zero tensile force on unloading.

desk verdict A sincere, modest phenomenological paper: the strain-dependent viscous prefactor is a real fix for the spurious tensile force, but the predictive claim is weakened by protocol-dependent fit parameters and an honest consistency check rather than a forward prediction. read the letter →

arxiv 2506.08855 v1 pith:6WJL7PYM submitted 2025-06-10 cond-mat.soft

Chandan Shakya , Luca Placidi , Anil Misra , Lars Kool , Anke Lindner , Jasper van der Gucht , Joshua A. Dijksman This is my paper
classification cond-mat.soft
keywords softgranularmaterialshydrogelpackingsviscoelasticitystandardlinearsolidmodelstrain-dependentviscositycyclicloadingno-tensionconditionrelaxationtimescales
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 proposes that the cyclic loading mechanics of cohesionless soft granular packings, such as compressed packs of hydrogel spheres or disks, can be captured by a single one-dimensional, one-timescale constitutive equation: a standard linear solid whose viscous branch carries a strain-dependent prefactor $(u_N/u_c)^b$. This prefactor switches off the viscous force as the contact displacement approaches zero, which prevents the tensile, adhesive forces that ordinary viscoelastic models produce on unloading. The model is fitted to strain-controlled compression cycles on 3D hydrogel sphere packings and to stress-controlled cycles on quasi-2D hydrogel disk packings, and it reproduces nonlinear force growth, relaxation at constant strain, recovery of peak force after waiting, and the absence of tensile force at zero strain. The authors position the model as a minimal description of time-history and rate effects, while noting that its parameters are not fully strain-rate independent and that high-rate and high-dissipation cases deviate, pointing to additional microscopic timescales.

What carries the argument

The central object is Eq. (3), the strain-dependent Maxwell element with a finite memory window. Its exponential kernel $\exp(w(\alpha-t+\tau)/\tau)$ weights recent displacement history over the interval $[t-\tau,t]$, with $w$ a constitutive weight chosen by the material rather than fixed to $e$ as in the Maxwell and standard linear solid kernels. The prefactor $(u_N/u_c)^b$ is the load-bearing innovation: $\tau$ sets the single relaxation timescale, $u_c$ sets the strain scale below which dissipation shuts off, and $b$ controls how sharply it shuts off. In the limit $\tau\to0$, Eq. (3) reduces to $c_N\dot u_N(u_N/u_c)^b$, recovering a Kelvin–Voigt-like rate response, so the model interpolates between rate-independent elastic and rate-dependent viscous behavior.

What would settle it

Take Protocol 6's highest tested strain rate, $\dot\epsilon=1.67\times10^{-2}\,\mathrm{s}^{-1}$, fix the parameters from a low-rate fit, and compare predicted versus measured peak force; the paper already reports the model's peak force keeps growing with rate while experiments plateau, so any extension where that plateau persists while the model continues to grow would falsify the single-timescale, rate-independent-parameter version.

Watch

Extended reading notes

Core claim

The central discovery is that the failure of standard viscoelastic models for soft granular packings, namely their prediction of tensile force when a cohesionless packing is unloaded to zero strain, can be cured by making the relaxation strength depend on the current strain. In the SDVES model, the total force is the sum of a nonlinear spring $F_{Ns}=k_N(u_N)^a$ and a Maxwell-type viscous element $$F_{NM}(t)=\frac{2c_N}{\$tau^{2}$}\left(\frac{u_N(t)}{u_c}\right)^b\int_{t-\tau}^{t}\bigl(u_N(t)-u_N(\$\alpha$)\bigr)\exp\!\left(w\frac{\$\alpha$-t+\tau}{\tau}\right)\,d\$\alpha$,$$ where the prefactor $(u_N/u_c)^b$ makes the viscous contribution negligible for $u_N\ll u_c$. With this correction, the model reproduces the qualitative force response across six strain-controlled protocols on 3D hydrogel packings and two stress-controlled protocols on 2D hydrogel disk packings, including nonlinear stiffening, relaxation, peak-force recovery after waiting, and zero tensile force at loss of contact. The paper also shows that standard Kelvin–Voigt, Maxwell, and standard linear solid models each fail on at least one of these features.

Load-bearing premise

The load-bearing premise is that one relaxation time $\tau$ and one scalar displacement $u_N$ are enough to encode the packing's entire deformation history; the paper's own fits require different parameters per protocol, so if multiple internal timescales are essential, the model's constants are not material constants.

Editorial extensions

If this is right

  • The SDVES model, with one set of parameters per protocol, captures the nonlinear compression curve, the relaxation plateau, the recovery of peak force after waiting longer than $\tau$, and zero tensile force on decompression for 3D hydrogel sphere packings.
  • The same constitutive form, using the measured strain as input, predicts the applied force in a quasi-2D stress-driven packing of chemically different PEGDA hydrogel disks, including the dissipation loop during slow oscillatory compression.
  • At high strain rates the model's peak force keeps rising whereas experiments flatten, so any complete description needs additional rate-dependent or multi-timescale viscous mechanisms.
  • The dissipated energy per cycle in both experiment and model scales as a power law in maximum strain, but the model's exponent and amplitude do not match the measured $\sim\epsilon^3$ scaling, so the decompression branch underestimates dissipation.
  • Because the fitted parameters $a$, $k_N$, $c_N$, and $u_c$ differ across protocols and vary with strain rate, the parameters are not true material constants; the model is a minimal interpolation tool rather than a predictive constitutive law for untested protocols.

Reading between the lines

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

  • If the strain-dependent prefactor is the essential mechanism, then a testable consequence is that the same model form should fit independent compression experiments on other cohesionless soft packings, such as emulsion or gel beads, with only $u_c$ and $b$ changed; the paper does not claim this.
  • The paper uses strain as the input even in the stress-controlled case, computing force as output. A true stress-driven prediction would require inverting Eq. (3) to give strain from an imposed force; one could test the model by solving this integral equation for $u_N(t)$ under a step stress and comparing with the measured creep curves.
  • The observed strain-rate dependence of the fitted exponent $a$ suggests that the model's elastic branch may itself need to be rate-dependent. A concrete extension would couple $a$ to $\dot u_N$ through a second timescale; the paper mentions multi-timescale improvements but does not propose this specific form.
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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

4 major / 5 minor

Summary. The paper introduces SDVES, a one-dimensional, single-relaxation-time constitutive model that modifies the standard linear solid model by adding a power-law elastic branch F_Ns = k_N u_N^a and a Maxwell-like viscous branch with a strain-dependent prefactor (u_N/u_c)^b. The prefactor is designed to suppress the viscous force at small displacements, which ensures that decompression to zero strain produces no tensile force. The model is fitted to six strain-controlled protocols on 3D hydrogel sphere packings and is also compared, by a consistency calculation, with stress-controlled 2D hydrogel disk experiments. The authors show that Kelvin-Voigt, Maxwell, and standard linear solid models fail to reproduce the observed nonlinear loading, relaxation, finite-time decompression, and absence of cohesion, and they report that SDVES qualitatively captures these features at low strain rates. The paper explicitly acknowledges rate-dependent fit parameters, the presence of multiple microscopic timescales, underestimation of dissipation, and the fact that inversion of the model for stress-controlled data is deferred.

Significance. The paper has several strengths: the model is simple and explicitly written down; the failure analysis of standard viscoelastic models is instructive; the experimental comparison covers two chemically and dimensionally different systems; and the authors are unusually candid about limitations, stating in Sec. IV that parameters 'should represent material constants, are not strain rate independent' and in Sec. V C that inverting the model is outside scope. If the model could be validated with fixed material parameters across protocols, it would be a useful minimal constitutive description for rate-dependent, cohesionless soft granular packings. However, the advertised predictive claim is not currently established: fits are performed per protocol, the no-tension property is built into the model by construction, and the 2D stress check is a closure test rather than a forward prediction. The paper is a reasonable descriptive-modeling contribution, but major revision is needed before the central claim can be accepted.

major comments (4)
  1. [Sec. IV, Table II] The central claim that SDVES 'captures the time-history and rate dependence' is supported only by per-protocol refits. For the same 3D hydrogel material, Table II changes a from 2.5 to 4, k_N from 0.39 M to 2.73 G, c_N from 400 to 900, and u_c from 2.6 mm to 5.3 mm across protocols, and Sec. IV explicitly states that 'the fit parameters, which should represent material constants, are not strain rate independent.' The one cross-protocol check shown in Fig. 5a (dotted line, parameters from Protocol 2 applied to Protocol 1) and the fixed-parameter check in Fig. 5d both deviate from the data, as the authors acknowledge. The agreement in Fig. 5 is therefore interpolation, not prediction. A fixed-parameter out-of-sample test, such as fitting Protocol 1 and then predicting Protocol 4 or Protocol 6 without refitting, is the missing experiment; without it, the abstract's predictive claim is not established.
  2. [Sec. III C, Eq. (3)] The absence of tensile force at decompression to zero strain is built into the model by construction, not an emergent prediction. The prefactor (u_N/u_c)^b is introduced in Eq. (3) precisely so that the viscous force becomes negligible for u_N << u_c, and Sec. III C states that 'we can choose parameters that ensure that forces are not attractive at the end of the decompression.' Consequently, the agreement at the end of Protocol 1 (Fig. 5a) is a check on the chosen functional form, not independent evidence that SDVES captures cohesionless behavior. The paper should either present this as a designed constraint or provide a test that could fail, for example by varying b, u_c, or the decompression rate through the crossover regime where the prefactor becomes significant.
  3. [Sec. V C, Fig. 6] The 2D stress-controlled comparison is a consistency check, not a forward prediction. The authors state that 'inverting the SDVES model to take imposed force as an input and predict the observed deformation ... is outside the scope of this research,' and that the measured strain is used as input with the model force output compared to the applied force. Thus Protocol 7 and Protocol 8 do not test whether SDVES predicts the strain response under imposed stress; they only test whether the model can close the loop when given the correct strain history. The conclusion that the model is 'effective ... for both strain and stress driven boundary conditions' overstates what Fig. 6 demonstrates.
  4. [Fig. 5, Sec. IV] The quality of the fits is not quantified, and known deviations affect the main claims. Fig. 5b shows deviations during the relaxation steps, Fig. 5g and 5h show systematic overestimation of force during decompression, and Fig. 5i shows that the model underestimates the dissipated work and gives an inconsistent exponent for the power-law scaling with strain amplitude. Without residuals, R^2, or parameter uncertainties, visual agreement in Fig. 5 is weak evidence given the model's seven free parameters. Reporting quantitative error measures, parameter uncertainties, and a simple identifiability check would substantially strengthen the paper.
minor comments (5)
  1. [Sec. IV] The text refers to 'the SDSL model' in the first paragraph; this should be SDVES, and 'Protocl 7' in Sec. V B should be 'Protocol 7.'
  2. [Sec. III C, Eq. (4)] In the sentence following Eq. (4), 'the exp(b) has been absorbed in the new material constant u_c prime' does not match the displayed change from Eq. (3) to Eq. (4), where the kernel changes from exp(w(alpha-t+tau)/tau) to exp((alpha-t)/tau); please clarify the algebra and the meaning of u_c prime.
  3. [Table II] The table header is missing a space ('... modelProtocol'), and the two rows both labeled Protocol 1 for Fig. 5a need a caption explaining which row corresponds to the solid and which to the dotted curve.
  4. [Fig. 5 caption] Panel (e) of Fig. 5 is described as a prediction for 'Protocol 5,' but the text compares it to Protocol 6 results; please correct the panel labeling.
  5. [Conclusions] The conclusion says the model 'works optimally at low strain rates,' which is more cautious than the abstract's claim that the model 'captures the time-history and rate dependence ... effectively'; align the wording or provide a qualification in the abstract.

Circularity Check

2 steps flagged · score 6.0 of 10

The advertised no-cohesion success is built into Eq. (3) by the (u_N/u_c)^b prefactor, and the rate/history 'capture' leans on per-protocol refits; fixed-parameter predictions are explicitly admitted to fail.

  1. self definitional [Sec. III C (Eq. 3 and surrounding text); abstract]
    "The term (u N (t)/uc)b in Eq. 3 is responsible for the 'strain-dependent' aspect in SDVES. ... if the displacement uN << uc is smaller than the characteristic one, then the viscous contribution to the response becomes negligible ... this combination of factors ensures that we can choose paramaters that ensure that forces are not attractive at the end of the decompression, consistent with our experimental data, and a feature that cannot be replicated from any of the other traditional models."

    The absence of cohesion at decompression is not a consequence derived from the model and then checked against data; it is inserted into the model. The prefactor (u_N/u_c)^b vanishes at u_N=0, so F_NM and F_Ns both vanish and the reaction is guaranteed to be non-attractive for all b>0, a>0. The abstract's claim that SDVES 'captures ... the necessary absence of cohesion' therefore restates the design input, not an independent prediction. The paper itself says 'The reason for such a factor is to avoid the negative force' in Sec. III C, confirming the feature is constructed rather than predicted.

  2. fitted input called prediction [Sec. IV and Table II (Appendix B)]
    "The parameters used for figure 5a is different from those used in 5b and c, even though the material used for the different protocols should be the same. ... However, if the parameters are modified, better fits can be achieved: the SDVES model works, yet the fit parameters, which should represent material constants, are not strain rate independent."

    Table II refits a, k_N, c_N, and u_c separately for each protocol on the same 3D hydrogel material (e.g., k_N ranges from 2.73G to 1.36M). With seven free parameters per protocol, the Fig. 5 'fit results' are interpolations within each fitted protocol rather than fixed-material forecasts. The abstract's central 'captures the time-history and rate dependence' is thereby supported by refitting freedom, not by a parameter-free prediction. The paper's own fixed-parameter attempt (Fig. 5d using 5a parameters at a different strain rate) is admitted to misalign, and only 'if the parameters are modified' does the model work, which is the fitted-input-called-prediction pattern.

full rationale

The model is not wholly circular: the memory-kernel form in Eq. (3) has genuine qualitative content, and Fig. 5e/f show fixed-parameter model responses across strain rates at low rates, with admitted failures at high rates and in dissipation scaling. However, the two load-bearing advertised successes are structurally weaker than claimed. First, the no-cohesion/no-tension property, highlighted as a feature 'that cannot be replicated from any of the other traditional models,' is guaranteed by the (u_N/u_c)^b prefactor introduced for exactly that purpose; it is a constraint built into Eq. (3), not a prediction confirmed by data. Second, the time-history and rate-dependence 'capture' is obtained with protocol-dependent parameters (Table II) that the paper concedes 'are not strain rate independent,' so the agreement in Fig. 5 is largely an interpolation. The 2D stress-controlled check in Sec. V C is also a consistency test using measured strain as input, with inversion deferred, so it does not independently validate predictive power under stress control. The paper is transparent about these limitations, which keeps this from being a deliberate concealment, but the logical structure still assigns to the model as a finding what was put in by construction. A fixed-parameter, out-of-sample test (one parameter set across all 3D protocols, and true stress-to-strain inversion in 2D) would be the missing step needed to turn these fits into predictions.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The model contributes a new phenomenological equation with four protocol-dependent fitted parameters (a, k_N, c_N, u_c) plus three chosen constants (tau, w, b). The key no-tension property follows from the ad hoc strain-dependent prefactor, so the model's success is partly a result of its construction. No new physical entities are postulated.

free parameters (7)
  • a (elastic non-linearity exponent) = 2.5, 3, 4 (per protocol, Table II)
    Controls non-linear elastic force F_Ns = k_N u_N^a. Fit to each protocol; varies with strain rate and protocol, so it is not a material constant.
  • k_N (elastic stiffness) = 0.39e6 to 2.73e9 N/m^a (Table II)
    Effective stiffness of the non-linear spring. Fit per protocol; spans orders of magnitude, indicating the model is not calibrated globally.
  • c_N (viscous strength) = 400 to 50,000 N s/m (Table II)
    Prefactor of the viscous force F_NM. Fit per protocol; varies widely, and the paper notes the parameters are not strain-rate independent.
  • u_c (characteristic displacement) = 0.15 to 5.3 mm (Table II)
    Scale that sets when the strain-dependent prefactor (u/u_c)^b becomes significant. Fit per protocol; chosen so the prefactor stays below unity.
  • tau (relaxation time) = 1500 s (3D), 50 s (2D)
    Sets the memory window and decay rate of the viscous kernel. Fixed per system, possibly informed by independent particle measurements in Ref. [21], but still a model input.
  • w (kernel weight exponent) = 6 (3D), 4 (2D)
    Constitutive weight factor in the exponential kernel exp(w(alpha - t + tau)/tau). Chosen by hand; no independent justification.
  • b (strain exponent in prefactor) = 2 (3D), 2.5 (2D)
    Exponent in the strain-dependent prefactor (u_N/u_c)^b. Chosen ad hoc to suppress the viscous force at small displacements, i.e., to prevent tensile force.
assumptions (4)
  • standard math Integral kernel representations of Maxwell and SLS models are valid (Appendix A, Eqs. A8, A15).
    The new model extends these solutions, so the background derivation of the exponential memory kernel is assumed standard.
  • domain assumption The packing response is dominated by a single relaxation time scale tau.
    The model uses one tau and one scalar displacement u_N. The paper itself notes (Sec. IV, Appendix B) that this is a limitation and that multiple timescales exist in the packings (Ref. [21]).
  • domain assumption The granular packings are cohesionless, so the force must be zero at zero imposed displacement.
    This is supported by the experiments (no tensile force observed), and it motivates the strain-dependent prefactor.
  • ad hoc to paper The strain-dependent prefactor (u_N/u_c)^b is an adequate representation of all non-linear and dissipation history effects.
    The prefactor is introduced ad hoc in Eq. (3) specifically to eliminate the negative force; it is not derived from micromechanics, and the paper provides no independent evidence for this form.

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Cite this review

Pith. "Pith review of Strain dependent viscous response describes the mechanics of cohesionless soft granular materials." pith.science (2026). https://pith.science/paper/6WJL7PYM

@misc{pith2026250608855,
  author       = {Pith},
  title        = {Pith review of: Strain dependent viscous response describes the mechanics of cohesionless soft granular materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WJL7PYM}},
  note         = {Machine review of arXiv:2506.08855}
}
read the original abstract

Granular materials are ubiquitous in nature and are used extensively in daily life and in industry. The modeling of these materials remains challenging; therefore, finding models with acceptable predictive accuracy that at the same time also reflect the complexity of the granular dynamics is a central research theme in the field. Soft particle packings present additional modeling challenges, as it has become clear that soft particles also have particle-level relaxation timescales that affect the packing behavior. We construct a simple one-dimensional, one-timescale model that replicates much of the essence of compressed hydrogel packing mechanics. We verify the model performance against both 3D and 2D packings of hydrogel particles, under both controlled strain and stress deformation conditions. We find that the modification of a Standard Linear Solid model with a strain dependent prefactor for the relaxation captures the time-history and rate dependence, as well as the necessary absence of cohesion effectively. We also indicate some directions of future improvement of the modeling.

Figures

Figures reproduced from arXiv: 2506.08855 by the authors.

Figure 1
Figure 1. FIG. 1. a) Schematic of the experimental setup b) picture of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Experimental results for various protocols described [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The weight of the past for two values of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Experimental force data (orange circles) and SDVES fit results (lines) for different experimental protocols described [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. a) Schematic overview of the microfluidic quasi-2D [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reference graph

Works this paper leans on

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    2) The response of the Maxwell (M) model to a constant non- null displacement asymptotically tends to zero

    The discontinuous response of the KV model (yellow). 2) The response of the Maxwell (M) model to a constant non- null displacement asymptotically tends to zero. (Purple) 3) The response of the Solid Linear Solid (SLS) model(Green) can be negative under null imposed displacement. none of the simpler models capture all the major char- acteristics of the com...

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    Here, the combination between elas- tic and viscous response is the series of a spring and a dashpot

    Maxwell (M) model Another model that is often used in viscoelasticity is the Maxwell model. Here, the combination between elas- tic and viscous response is the series of a spring and a dashpot. Thus, the relative displacementu M between 11 the two sides of the Maxwell element is the sum between those of the springu M sand of the dashpotu M d. uM =u M s+u ...

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    Kelvin–V oigt (KV) model In a Kelvin–Voigt model the combination between elas- tic and viscous response is the parallel of a spring and a dashpot. The reaction forceF KV is the sum of that of the spring, e.g. with stiffnessk KV , and of the dashpot, e.g. with viscosityc KV , FKV =k KV uKV +c KV ˙uKV .(A3) The KV model used a spring linear with displacemen...

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    Standard Linear Solid(SLS) model Another model that is often used in viscoelasticity is the standard linear solid model. Here, the combination between elastic and viscous response is the parallel of a spring and a Maxwell element. Thus, the relative dis- placementu st between the two sides of the standard el- ement is the same for the spring, e.g. with st...

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