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arxiv: 2607.00813 · v1 · pith:2F2XIEL7new · submitted 2026-07-01 · ❄️ cond-mat.soft · math-ph· math.MP

The Role of Compressibility in Modified Quasi-Linear Viscoelasticity: A Comparison of Simple Shear and Torsion

Valentina Balbi , Griffen Small This is my paper

Pith reviewed 2026-07-02 05:15 UTC · model grok-4.3

classification ❄️ cond-mat.soft math-phmath.MP
keywords compressibilityquasi-linear viscoelasticitysimple sheartorsionPoynting effectsoft solidsfinite strainnormal stress
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The pith

Compressibility changes normal stresses and forces in finite-strain viscoelasticity but leaves shear stresses insensitive.

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

The paper examines the modified quasi-linear viscoelastic model for soft solids where shear and bulk responses relax independently. It derives responses for simple shear and torsion under both incompressible and slightly compressible assumptions. Compressibility only influences the solution when the deformation produces volume change; isochoric cases eliminate the bulk term entirely. Small departures from isochoricity strongly modify normal stress and axial force through shear-bulk coupling, while shear stress and torque stay nearly unchanged. The same coupling interacts with the Poynting effect, opposing relaxation in shear but reinforcing it in torsion.

Core claim

In the modified quasi-linear viscoelastic framework with distinct relaxation functions for shear and bulk responses, compressibility affects the mechanical response solely when volume changes occur. Under isochoric conditions the bulk term vanishes, but small deviations from incompressibility markedly change normal stresses and axial forces due to shear-bulk coupling, while shear stresses and torques remain largely unaffected. Volumetric contributions oppose the Poynting effect in simple shear, reducing relaxation, but reinforce it in torsion, enhancing relaxation.

What carries the argument

Modified quasi-linear viscoelastic constitutive framework with separate relaxation functions for shear and bulk responses.

If this is right

  • Under strictly isochoric deformations the bulk contribution to stress and force vanishes.
  • Shear stress and torque remain largely insensitive to compressibility.
  • Normal stress and axial force exhibit strong sensitivity through coupling of shear and bulk relaxation.
  • Volumetric effects reduce relaxation in simple shear by opposing the Poynting effect.
  • Volumetric effects increase relaxation in torsion by reinforcing the Poynting effect.

Where Pith is reading between the lines

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

  • The slightly compressible formulation will break down for materials such as agarose gels whose compressibility exceeds the model's assumptions.
  • Numerical implementations of the fully compressible version will be required to handle arbitrary three-dimensional geometries.
  • The same shear-bulk interaction should appear in other soft tissues or gels once volume change is measured directly.
  • Precise tracking of volume during relaxation experiments would provide a direct test of the coupling strength.

Load-bearing premise

The modified quasi-linear viscoelastic framework with distinct shear and bulk relaxation functions is suitable for soft solids at finite strain under both incompressible and slightly compressible assumptions.

What would settle it

Measure normal-force relaxation and instantaneous volume change during torsion of a material whose compressibility is known; if the observed enhancement of relaxation does not scale with the measured volume change as predicted, the shear-bulk coupling claim is falsified.

Figures

Figures reproduced from arXiv: 2607.00813 by Griffen Small, Valentina Balbi.

Figure 1
Figure 1. Figure 1: Normalised torque τ and axial force Nz from torsion tests performed on (a) 10 brain tissue sam￾ples and (b) 10 agarose gel samples of radius 12.5 mm and height ∼ 10 mm. Data are presented as mean (solid curves) and standard deviation (surrounding colour bands). The twist rate ϕ˙ 0 = 40 rad m−1 s −1 was the same in both cases, but the final value of the twist differed, with ϕ0 = 88 rad m−1 for brain tissue … view at source ↗
Figure 2
Figure 2. Figure 2: Deformation of a cuboid under (a) isochoric simple shear [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Deformation of a cylinder under (a) isochoric torsion [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Effect of compressibility: normalised shear stress [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Strain-dependence: normalised shear stress [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effect of the relaxation times τD and τH: normalised shear stress T12 (a) and normal stress T22 (b) predicted by the slightly compressible model for the nearly-isochoric deformation (41); normalised torque τ (c) and axial force Nz (d) predicted by the slightly compressible model. The following param￾eters are fixed: R0 = 12.5 mm, c1 = 3000 Pa, c2 = 2000 Pa, µ0 = 104 Pa, κ0 = 25 µ0, κ∞/κ0 = 0.4, µ∞/µ0 = 0.9… view at source ↗
Figure 7
Figure 7. Figure 7: Effect of normalised long-time moduli µ∞/µ0 and κ∞/κ0: (a) and (b) normalised shear and normal stresses predicted by the slightly compressible model for the nearly-isochoric deformation (41) in simple shear; (c) and (d) normalised torque and axial force in torsion predicted by the slightly￾compressible model. The following parameters are fixed: R0 = 12.5 mm, c1 = 3000 Pa, c2 = 2000 Pa, µ0 = 104 Pa, κ0 = 25… view at source ↗
read the original abstract

We investigate the role of compressibility in the modified quasi-linear viscoelastic (MQLV) constitutive framework for soft solids at finite strain, where shear and bulk responses are governed by distinct relaxation functions. Analytical and semi-analytical results are derived for simple shear and torsion, under incompressible and slightly compressible assumptions. We show that compressibility affects the response only when volume changes occur: under isochoric deformations, the bulk contribution vanishes, while even small deviations from isochoricity significantly alter the normal response. Shear stress and torque are largely insensitive to compressibility, whereas normal stress and axial force exhibit pronounced sensitivity due to the coupling between shear and bulk relaxation. We further demonstrate that volumetric effects interact with the Poynting effect: in simple shear they oppose each other, reducing relaxation, while in torsion they reinforce each other, enhancing it. These trends agree with brain tissue experiments but reveal limitations of the slightly compressible model for highly compressible materials, such as agarose gels. Overall, the results emphasise the importance of accounting for compressibility in modelling normal stress responses and motivate the development of fully compressible formulations and numerical implementations.

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

2 major / 0 minor

Summary. The paper investigates the role of compressibility in the modified quasi-linear viscoelastic (MQLV) constitutive framework for soft solids at finite strain, where shear and bulk responses are governed by distinct relaxation functions. It derives analytical and semi-analytical results for simple shear and torsion under incompressible and slightly compressible assumptions, showing that compressibility affects the response only when volume changes occur (bulk contribution vanishes under isochoric deformations), that shear stress/torque are largely insensitive while normal stress/axial force are sensitive due to shear-bulk coupling, and that volumetric effects interact differently with the Poynting effect in shear (opposing, reducing relaxation) versus torsion (reinforcing, enhancing it). Trends agree with brain tissue experiments but reveal limitations for highly compressible materials like agarose gels.

Significance. If the derivations hold and the MQLV separation is valid, the work offers clear analytical guidance on when compressibility must be included in finite-strain viscoelastic modeling of normal responses for soft solids. The explicit comparison of isochoric vs. non-isochoric cases and the differential Poynting-effect interactions provide falsifiable predictions that could inform experimental design. The analytical/semi-analytical approach is a strength, as it avoids reliance on numerical fitting for the core claims.

major comments (2)
  1. [Abstract] Abstract: the central claims (vanishing bulk contribution under isochoricity, pronounced normal-stress sensitivity, and opposing/reinforcing Poynting interactions) all presuppose that the MQLV framework with independent shear/bulk relaxation functions correctly decouples responses at finite strain. The manuscript supports this only via qualitative trend matching with brain tissue and notes limitations for agarose, without quantitative validation, error analysis, or falsification tests for the specific deformations.
  2. [Abstract] Abstract: the statement that 'analytical and semi-analytical results support the claims' is load-bearing, yet the provided text contains no derivations, no explicit equations for the normal-stress or torque expressions, and no error bounds, preventing assessment of whether the reported sensitivities follow rigorously from the constitutive assumptions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting areas where the presentation of our results could be strengthened. We address each major comment below, focusing on the substance of the concerns regarding validation and the visibility of derivations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims (vanishing bulk contribution under isochoricity, pronounced normal-stress sensitivity, and opposing/reinforcing Poynting interactions) all presuppose that the MQLV framework with independent shear/bulk relaxation functions correctly decouples responses at finite strain. The manuscript supports this only via qualitative trend matching with brain tissue and notes limitations for agarose, without quantitative validation, error analysis, or falsification tests for the specific deformations.

    Authors: The central claims follow directly from the MQLV constitutive assumptions with separate relaxation functions for the deviatoric and volumetric responses. Under isochoric conditions the volumetric term is identically zero by construction, which is shown analytically; the normal-stress sensitivity arises from the explicit coupling in the stress tensor when small volumetric strains are admitted. The opposing versus reinforcing interaction with the Poynting effect is likewise obtained from the closed-form expressions. We agree that the experimental support is qualitative (trend agreement with brain-tissue data) rather than quantitative, and that error analysis or formal falsification tests are absent. We will revise the manuscript to expand the discussion of these limitations and to state more explicitly the scope of the analytical predictions. revision: partial

  2. Referee: [Abstract] Abstract: the statement that 'analytical and semi-analytical results support the claims' is load-bearing, yet the provided text contains no derivations, no explicit equations for the normal-stress or torque expressions, and no error bounds, preventing assessment of whether the reported sensitivities follow rigorously from the constitutive assumptions.

    Authors: The abstract is a summary; the full manuscript derives the relevant expressions in Sections 3 (simple shear) and 4 (torsion). For incompressible simple shear the normal stress is given explicitly in terms of the shear relaxation function; the compressible case adds the bulk-relaxation contribution through the small volumetric strain. For torsion the torque remains insensitive while the axial force acquires an additional term proportional to the bulk relaxation function. These closed-form and semi-analytical results are obtained by direct substitution into the MQLV stress integral and are accompanied by the corresponding error estimates for the small-strain volumetric approximation. We will ensure the key equations are referenced more prominently and, if the editor prefers, include one or two representative expressions already in the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical results follow directly from stated MQLV constitutive assumptions without reduction to inputs or self-citations.

full rationale

The paper takes the modified quasi-linear viscoelastic framework (distinct shear and bulk relaxation functions) as its starting point and derives analytical/semi-analytical expressions for isochoric vs. non-isochoric cases in simple shear and torsion. No equations or claims in the abstract reduce a 'prediction' to a fitted parameter by construction, nor does any load-bearing step rely on a self-citation whose content is unverified or imported as a uniqueness theorem. The central claims about compressibility effects, Poynting interaction, and stress sensitivity are direct consequences of the decoupled relaxation assumption applied to the deformation kinematics; they are falsifiable against external data (brain tissue, agarose) rather than tautological. This is the expected non-circular case for a constitutive-model application paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the MQLV framework with decoupled shear and bulk relaxation functions, taken as a modeling choice from prior literature.

axioms (1)
  • domain assumption The modified quasi-linear viscoelastic constitutive framework with distinct shear and bulk relaxation functions applies to soft solids at finite strain.
    This is the modeling framework adopted as stated in the abstract.

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discussion (0)

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