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arxiv: 2604.03467 · v3 · pith:NRUJV34Jnew · submitted 2026-04-03 · ⚛️ physics.flu-dyn · cond-mat.soft

A Solid-Based Approach for Modeling Simple Yield-Stress Fluids: Rheological Transitions, Overshoot and Relaxation

Jehyeok Choi , Ju Min Kim , Kwang Soo Cho This is my paper

Pith reviewed 2026-05-21 09:51 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.soft
keywords yield-stress fluidsconstitutive equationstress overshootviscoelastic solidrheological transitionsnormal stress differenceplastic responsestart-up shear
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The pith

A viscoelastic solid model for yield-stress fluids explains stress overshoot through normal stress coupling in homogeneous flow.

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

The paper proposes a new constitutive equation for simple yield-stress fluids based on a Zener-type viscoelastic solid with an added dashpot and nonlinear viscosity. This model is designed to capture both steady and transient behaviors like start-up shear, creep, and stress relaxation without including thixotropy. A key finding is that the model accurately predicts the observed stress overshoot during start-up shear, attributing it to a mechanism where normal stress differences increase the stress invariant and speed up the plastic response. This homogeneous explanation contrasts with ideas involving hardening or microstructural heterogeneity. If valid, it offers a simpler way to model these fluids' dynamics in applications like battery slurries and ink writing.

Core claim

The proposed constitutive equation, derived from a Zener-type viscoelastic solid element combined with a parallel linear dashpot, nonlinear viscosity model, flow rule, back stress evolution, and Kroner-Lee decomposition, reproduces the rheological characteristics of simple yield-stress fluids. It accurately predicts stress overshoot in start-up shear, which originates from a homogeneous mechanism in which normal stress difference enhances the stress invariant and thereby accelerates the plastic response, rather than from isotropic hardening or spatially heterogeneous microstructural evolution.

What carries the argument

The Zener-type viscoelastic solid element with parallel dashpot and nonlinear viscosity, incorporating the Kroner-Lee decomposition to ensure frame invariance, which allows the normal stress differences to couple into the stress invariant driving plastic flow.

If this is right

  • The model reproduces start-up shear, creep, and stress relaxation behaviors in a manner consistent with experiments on simple yield-stress fluids.
  • Stress overshoot arises specifically from normal stress difference enhancing the stress invariant to accelerate plastic response.
  • The explanation relies on homogeneous mechanisms inside the constitutive equations instead of isotropic hardening or heterogeneous evolution.
  • The combination satisfies material frame invariance while describing transient rheological transitions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This homogeneous coupling could simplify flow predictions in applications like electrode slurries where spatial resolution is limited.
  • Numerical implementations of the model in non-uniform geometries might reveal whether the overshoot mechanism remains dominant.
  • The approach suggests that other transient features in yield-stress fluids could be captured by extending similar solid-based elements without adding thixotropy.
  • Direct comparison with particle-level simulations could isolate the contribution of normal stress terms from any microstructural effects.

Load-bearing premise

The overshoot is assumed to come only from the homogeneous normal-stress coupling in the equations rather than any unmodeled spatial variations or structure breakdown.

What would settle it

Spatially resolved measurements or simulations during start-up shear that show no significant heterogeneity while still observing the overshoot would support the claim, whereas evidence of strong local variations would challenge it.

Figures

Figures reproduced from arXiv: 2604.03467 by Jehyeok Choi, Ju Min Kim, Kwang Soo Cho.

Figure 1
Figure 1. Figure 1: (a) Carbopol microstructure schematic: jammed swollen PAA microgels immersed in interstitial solvent. (b) 1D mechanical analogue: Zener-type viscoelastic solid in parallel with a solvent dashpot. The strain of Spring 1 is selected as the elastic strain e  , whereas that of the Kelvin-Voigt model is regarded as the plastic strain p  . The sum of these strains is the total strain: . e p      (1) The e… view at source ↗
Figure 3
Figure 3. Figure 3: Steady state stress of Carbopol 940 0.2 wt% solution as a function of shear rate obtained from steady shear test. Black circle represents experimental data; red line represents best fit of Herschel-Bulkley equation for   1 s ; and blue and green lines represent fitting results using Eyring and Carreau-Yasuda viscosities in the general Herschel-Bulkley equation, respectively [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 4
Figure 4. Figure 4: Shear stresses of start-up shear calculated via numerical integration of Eq. (14) with (a) Eyring viscosity and (b) Carreau-Yasuda viscosity models. The parameters are set as follows:  B 0.1 , G  20 Pa , GB 100Pa ,  o 30Pa ,    o 70Pa s , 1Pa s s    ,   2.5 ,   1.5 and     o o [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Shear-stress relaxation as a function of dimensionless time in stress￾relaxation tests obtained from (a) Eyring viscosity and (b) Carreau–Yasuda viscosity models. The parameters are set as follows: t o o   20  , tmax o   40  ,  B 0.1 , G  20 Pa , GB 100Pa ,  o 30Pa ,    o 70Pa s , 1Pa s s    ,   2.5 ,   1.5 and     o o [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Shear stresses of start-up shear calculated via numerical integration of Eq. (68) using (a) Eyring viscosity and (b) Carreau–Yasuda viscosity. The parameters are set as follows:  B 0.1, G  20 Pa , GB 100Pa ,  o 30Pa ,    o 70Pa s , 1Pa s s    ,   2.5 ,   1.5 and     o o [PITH_FULL_IMAGE:figures/full_fig_p031_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Steady state stresses calculated using (a) Eyring viscosity and (b) Carreau￾Yasuda viscosity plotted as a function of normalized shear rate o o o     . Open circles represent steady state stresses obtained from numerical integration of Eq. (68); red lines represent regression results of steady-state stress obtained using Herschel–Bulkley equation (HB regression); blue lines represent regression results… view at source ↗
Figure 9
Figure 9. Figure 9: Shear stresses of stress relaxation calculated using (a) Eyring viscosity and (b) Carreau-Yasuda viscosity. The parameters are set as follows: t o o   20  , tmax o   40  ,  B 0.1 , G  20 Pa , GB 100Pa ,  o 30Pa ,    o 70Pa s , 1Pa s s    ,   2.5 ,   1.5 and     o o [PITH_FULL_IMAGE:figures/full_fig_p035_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Numerical integration results of Eq. (68) for creep test. Strains calculated using (a) Eyring viscosity and (b) Carreau-Yasuda viscosity and shear rates calculated using (c) Eyring viscosity and (d) Carreau-Yasuda viscosity. In order to facilitate understanding, the strains are normalized to their values at 2 t 10   . The parameters are set as follows:  B 0.1 , G  20 Pa , GB 100Pa ,  o 30Pa ,  … view at source ↗
read the original abstract

Yield-stress fluids are ubiquitous and encountered in diverse fields ranging from natural muddy flows to industrial applications such as secondary battery electrode slurries and direct ink writing. Despite the proposal of various constitutive equations, few models have been shown to successfully predict both steady and transient rheological behaviors in yield-stress fluids. In this study, a constitutive equation is hereby proposed, offering a comprehensive description of the rheological characteristics observed in simple yield-stress fluids, excluding thixotropy, such as the Carbopol dispersion. The constitutive equation is derived from a Zener-type viscoelastic solid element combined with an additional linear dashpot connected in parallel, together with a nonlinear viscosity model, a flow rule, an evolution equation for the back stress, and the Kroner-Lee decomposition. This combination satisfies the principle of material frame invariance. The proposed model successfully reproduces the rheological characteristics qualitatively in a manner consistent with experimental observations conducted during start-up shear, creep, and stress relaxation tests. In particular, the present viscoelastic solid-based constitutive equation is shown to accurately predict stress overshoot during start-up shear. Importantly, the overshoot is found to originate from a homogeneous mechanism in which normal stress difference enhances the stress invariant and thereby accelerates the plastic response, rather than from isotropic hardening or spatially heterogeneous microstructural evolution. This study is expected to facilitate a deeper understanding of the intricate dynamics governing the flow of yield-stress fluids.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a constitutive model for simple yield-stress fluids (e.g., Carbopol) based on a Zener viscoelastic solid element in parallel with a linear dashpot, augmented by a nonlinear viscosity function, back-stress evolution, a flow rule, and Kroner-Lee decomposition to ensure frame invariance. It claims qualitative reproduction of experimental trends in start-up shear, creep, and stress relaxation, with the key mechanistic finding that stress overshoot during start-up shear arises from a homogeneous process in which normal-stress differences enhance the stress invariant and accelerate plastic flow, rather than from isotropic hardening or spatial heterogeneity.

Significance. If the central claims hold, the solid-based formulation provides a compact way to capture both steady and transient rheology of simple yield-stress fluids without explicit thixotropy. The proposed homogeneous mechanism for overshoot supplies a falsifiable, local explanation that could be tested in other geometries and materials, potentially informing modeling of industrial flows such as battery slurries and direct-ink writing.

major comments (3)
  1. [Abstract] Abstract: the assertion that overshoot 'originates from a homogeneous mechanism in which normal stress difference enhances the stress invariant' is presented as a central result, yet the manuscript supplies no spatially resolved velocity or strain-field measurements during the overshoot window and performs no comparison against a heterogeneous or thixotropic alternative that could produce similar bulk curves.
  2. [Results] Results (start-up shear section): reproduction of experimental overshoot is described only qualitatively; no error metrics, parameter tables, or discussion of how many free parameters in the nonlinear viscosity and back-stress evolution were adjusted to match the data are provided, leaving the predictive power and degree of fitting unclear.
  3. [Model formulation] Model formulation: the constitutive equations are strictly local and therefore presuppose homogeneity; without an explicit check that the experimental flow remains uniform (or a side-by-side heterogeneous simulation), the distinction between the proposed normal-stress mechanism and possible unmodeled spatial effects cannot be established.
minor comments (2)
  1. [Abstract] The abstract states both 'qualitatively' and 'accurately predict'; this wording inconsistency should be resolved.
  2. [Appendix or supplementary material] No table listing the numerical values of the free parameters or their sensitivity is included, which would aid reproducibility.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the scope and limitations of our modeling study. We address each major point below and indicate planned revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that overshoot 'originates from a homogeneous mechanism in which normal stress difference enhances the stress invariant' is presented as a central result, yet the manuscript supplies no spatially resolved velocity or strain-field measurements during the overshoot window and performs no comparison against a heterogeneous or thixotropic alternative that could produce similar bulk curves.

    Authors: We agree that the manuscript presents no new spatially resolved data and does not perform side-by-side comparisons with heterogeneous or thixotropic models. The central claim is a mechanistic prediction obtained from the local constitutive equations under the standard assumption of homogeneous simple shear. We will revise the abstract to qualify the statement as a result within the homogeneous modeling framework and add a sentence in the discussion section noting that local flow visualization would be required to test the mechanism experimentally. Direct comparison to alternative models is outside the present scope, which focuses on the solid-based formulation for non-thixotropic fluids. revision: partial

  2. Referee: [Results] Results (start-up shear section): reproduction of experimental overshoot is described only qualitatively; no error metrics, parameter tables, or discussion of how many free parameters in the nonlinear viscosity and back-stress evolution were adjusted to match the data are provided, leaving the predictive power and degree of fitting unclear.

    Authors: We accept that the current presentation is qualitative and lacks quantitative assessment. In the revised manuscript we will insert a parameter table listing all values used for the start-up shear simulations, indicate which parameters were fitted versus held fixed from independent tests, and report simple error metrics (e.g., relative deviation of the overshoot peak stress and steady-state viscosity from the cited experimental curves). These additions will make the extent of fitting explicit. revision: yes

  3. Referee: [Model formulation] Model formulation: the constitutive equations are strictly local and therefore presuppose homogeneity; without an explicit check that the experimental flow remains uniform (or a side-by-side heterogeneous simulation), the distinction between the proposed normal-stress mechanism and possible unmodeled spatial effects cannot be established.

    Authors: The equations are written as a local constitutive model, which is the conventional approach for point-wise material response in continuum rheology. We will add a dedicated paragraph in the model-formulation section that states the homogeneity assumption, cites literature indicating that Carbopol dispersions exhibit largely uniform flow in simple shear after yielding, and notes that the normal-stress mechanism operates at the material-point level. A coupled heterogeneous simulation lies beyond the scope of the present work, which centers on constitutive behavior rather than full-field flow computation. revision: partial

standing simulated objections not resolved
  • New spatially resolved velocity or strain-field measurements during the overshoot window, which would require experimental work outside the modeling focus of this study.

Circularity Check

1 steps flagged

Overshoot mechanism claim reduces to homogeneous model assumptions by construction

specific steps
  1. fitted input called prediction [Abstract]
    "the present viscoelastic solid-based constitutive equation is shown to accurately predict stress overshoot during start-up shear. Importantly, the overshoot is found to originate from a homogeneous mechanism in which normal stress difference enhances the stress invariant and thereby accelerates the plastic response, rather than from isotropic hardening or spatially heterogeneous microstructural evolution."

    The model is formulated as a local (pointwise) set of ODEs that enforce homogeneity by construction. Parameters are chosen so the equations match the same start-up shear overshoot data. The stated origin of the overshoot is therefore generated inside the fitted homogeneous equations rather than demonstrated by spatially resolved measurements or comparison to heterogeneous alternatives.

full rationale

The paper constructs a local constitutive model (Zener solid + parallel dashpot, nonlinear viscosity, back-stress evolution, Kroner-Lee decomposition) that assumes spatial homogeneity from the outset. Parameters are selected to reproduce bulk start-up shear curves that exhibit overshoot. The central claim that overshoot originates specifically from normal-stress enhancement of the stress invariant is obtained by solving these same equations; the model cannot generate or test spatial heterogeneity, so the mechanistic attribution is a direct output of the fitted homogeneous framework rather than an independent verification. This produces moderate circularity in the interpretation of the result even though the equations themselves are physically motivated.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard continuum-mechanics assumptions plus several fitted functions whose values are not supplied in the abstract.

free parameters (2)
  • nonlinear viscosity function parameters
    Coefficients in the stress-dependent viscosity that control the rate of plastic flow are adjusted to match experimental overshoot and relaxation curves.
  • back-stress evolution constants
    Parameters governing the rate of back-stress relaxation are chosen to reproduce creep and stress-decay data.
axioms (2)
  • standard math Material frame invariance is satisfied by the Kroner-Lee decomposition and objective rates.
    Invoked to ensure the constitutive equations remain valid under rigid rotations.
  • domain assumption The fluid is simple (non-thixotropic); structure evolution is absent.
    Explicitly stated as the scope limitation for Carbopol-like dispersions.

pith-pipeline@v0.9.0 · 5785 in / 1494 out tokens · 43640 ms · 2026-05-21T09:51:02.394194+00:00 · methodology

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