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arxiv: 2506.16785 · v3 · submitted 2025-06-20 · 🧮 math-ph · math.MP

A few notes about viscoplastic rheologies

Tom\'a\v{s} Roub\'i\v{c}ek This is my paper

Pith reviewed 2026-05-19 08:21 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords viscoplastic rheologiesconvex analysisdissipation potentialserial and parallel combinationsgeomaterialsharmonic meansnonlinear viscosities
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The pith

Convex analysis synthesizes a single convex viscoplastic dissipation potential by combining viscous and plastic elements in serial and parallel.

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

The paper uses rigorous tools of convex analysis to examine serial and parallel combinations of linear viscosity and perfect plasticity, including nonlinear viscosities. It aims to create a unified convex dissipation potential for viscoplastic behavior from the individual potentials of these elements. A reader would care because this provides a mathematical basis for models applied to geomaterials and allows direct comparison with common empirical approximations using harmonic means. This could lead to more consistent and accurate rheological models without ad hoc adjustments.

Core claim

The rigorous tools of convex analysis are used to examine various serial and parallel combinations of linear viscosity and perfect plasticity. Nonlinear viscosities are also considered. The general aim is to synthesize a single convex viscoplastic dissipation potential from the potentials of particular viscous or plastic elements. Rigorous serial-viscosity models are then compared with empirical models based on harmonic means, which are commonly used for various geomaterials.

What carries the argument

A single convex viscoplastic dissipation potential constructed via serial and parallel combination rules from individual viscous and plastic potentials.

Load-bearing premise

That the rules for serial and parallel combinations preserve convexity of the dissipation potential and that the resulting models sufficiently correspond to physical behavior in geomaterials.

What would settle it

Direct comparison of stress-strain curves predicted by the convex model against experimental data from geomaterials where the harmonic mean approximation shows clear discrepancies.

Figures

Figures reproduced from arXiv: 2506.16785 by Tom\'a\v{s} Roub\'i\v{c}ek.

Figure 1
Figure 1. Figure 1: Two options in combination of a viscous damper with a perfect plasticity with the activation threshold σa . 2 Viscoplasticity: basic scenarios First, we consider the perfect plasticity governed by the convex homogeneous degree-1 (non￾smooth) potential ζ1(·) = σa | · | with σa > 0 and an activation (yield) stress in Pa=J/m3 and a linear viscosity governed by the homogeneous degree-2 (here quadratic) potenti… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic illustration of the convex nonsmooth dissipation potential ζvp from (2) with both D2 > 0 and σa > 0, its subdifferential ∂ζvp which is set-valued at 0, and its (smooth) conjugate ζ ∗ vp and its derivative (= single-valued inverse of ∂ζvp) used to model (rate-dependent) plasticity; cf. also [23, Chap.27] [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of the convex smooth dissipation potential ζvp from (3) with both D2 > 0 and σa > 0, its continuous differential ζ ′ vp, and its (nonsmooth) conjugate ζ ∗ vp and its derivative (= set-valued inverse of ζ ′ vp) used to model plasticity combined with a creep. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two (mutually equivalent) variants of a combination of two viscous dampers with a perfect plasticity with the activation thresholds σa and σea leading to a continuous ζ ∗ vp ′ . ζ1□ ζ2 can be not trivial in particular cases. Interestingly, for the quadratic function ζ2 as in (2), ζ1□ ζ2 is the Yosida approximation of ζ1 , namely Y1/Dζ1, cf. (33) below. In this case, ζvp is smooth (continuously differentiab… view at source ↗
Figure 5
Figure 5. Figure 5: Schematic illustration of the bi-visco-plastic model from from [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A comparison of the serial arrangement of the linear viscous (i.e. diffusion) creep and the power-law Norton-Hoff model (17) with n = 2, n = 3 (i.e. the dislocation creep), and n = ∞ (i.e. the perfect plasticity as the min-formula in (4)). Both the effective viscosity (left) and the corresponding stress (right) depending on the strain rate ε are displayed. Also the harmonically-averaged empirical model (24… view at source ↗
Figure 7
Figure 7. Figure 7: A schematic 1-dimensional illustration of a generalized anti-Zener (also called [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

The rigorous tools of convex analysis are used to examine various serial and parallel combinations of linear viscosity and perfect plasticity. Nonlinear viscosities are also considered. The general aim is to synthesize a single convex ``viscoplastic'' dissipation potential from the potentials of particular viscous or plastic elements. Rigorous serial-viscosity models are then compared with empirical models based on harmonic means, which are commonly used for various geomaterials.

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

1 major / 2 minor

Summary. The manuscript uses tools from convex analysis to derive single convex viscoplastic dissipation potentials by combining the potentials of individual viscous and plastic rheological elements under serial and parallel arrangements. It treats both linear viscosity/perfect plasticity and nonlinear viscosity cases, then compares the resulting rigorous serial-viscosity models against empirical harmonic-mean approximations commonly employed for geomaterials.

Significance. If the central constructions hold, the work supplies a systematic, convexity-preserving route to synthesizing viscoplastic potentials from elementary components. This strengthens the theoretical underpinnings of rheological modeling in geomechanics by replacing ad-hoc combinations with operations (sums and infimal convolutions) whose properties are guaranteed by convex analysis, while the comparison to harmonic-mean models offers a useful consistency check against existing empirical practice.

major comments (1)
  1. [§3 and abstract] The abstract and §3 state that serial-viscosity models are compared with harmonic-mean empirical models as a consistency check. However, the manuscript provides no quantitative assessment (e.g., relative error or parameter-range plots) of how closely the convex-analysis result matches the harmonic-mean form for representative nonlinear viscosity exponents; without this, the practical utility of the rigorous model as a replacement or refinement remains difficult to judge.
minor comments (2)
  1. [Throughout] The notation for dissipation potentials (e.g., the distinction between viscous, plastic, and combined potentials) would benefit from an explicit symbol table or consistent subscript convention to avoid ambiguity when multiple elements are combined.
  2. [§2] A brief remark on the physical interpretation of the resulting combined potentials (e.g., effective yield stress or effective viscosity) would help readers connect the mathematical constructions to measurable material properties.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [§3 and abstract] The abstract and §3 state that serial-viscosity models are compared with harmonic-mean empirical models as a consistency check. However, the manuscript provides no quantitative assessment (e.g., relative error or parameter-range plots) of how closely the convex-analysis result matches the harmonic-mean form for representative nonlinear viscosity exponents; without this, the practical utility of the rigorous model as a replacement or refinement remains difficult to judge.

    Authors: We agree that a quantitative assessment would strengthen the presentation and help readers evaluate the practical implications. In the revised manuscript we will add figures in §3 showing the relative difference (and, where appropriate, absolute deviation) between the rigorous serial-viscosity potential obtained via infimal convolution and the harmonic-mean approximation, for representative nonlinear viscosity exponents (e.g., power-law indices n=1.5, 2 and 3) over a suitable range of normalized stress or strain-rate values. These plots will be accompanied by a brief discussion of the observed agreement and any systematic deviations. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies standard convex-analysis operations (sums for parallel combinations and infimal convolutions for serial combinations) to synthesize a single convex viscoplastic dissipation potential from component potentials. These operations are established mathematical facts that preserve convexity independently of the specific models under consideration. The comparison to empirical harmonic-mean approximations is presented explicitly as a consistency check rather than an equivalence or prediction derived from the synthesis itself. No load-bearing step reduces to a self-definition, a fitted parameter renamed as a prediction, or a self-citation chain that substitutes for independent justification. The derivation remains self-contained against external convex-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions from convex analysis and continuum mechanics without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption Dissipation potentials of individual viscous and plastic elements are convex.
    Standard premise in convex-analysis treatments of dissipative systems, invoked to ensure the combined potential remains convex.
  • domain assumption Serial and parallel combinations are realized through infimal convolution or supremal convolution operations.
    Common construction in rheological modeling that the abstract relies upon to synthesize the overall potential.

pith-pipeline@v0.9.0 · 5584 in / 1343 out tokens · 52567 ms · 2026-05-19T08:21:38.440458+00:00 · methodology

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