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arxiv: 2604.16072 · v1 · submitted 2026-04-17 · 🧮 math-ph · math.MP

Identification of optimal history variables and corresponding hereditary laws in linear viscoelasticity

Ignacio Romero , Michael Ortiz This is my paper

Pith reviewed 2026-05-10 07:50 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords linear viscoelasticityhereditary lawsinternal variableshistory operatorcompact operatorsKolmogorov N-widthsreduced-order modelingthermodynamic consistency
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The pith

The history operator in linear viscoelasticity is compact, admitting optimal finite-rank internal-variable approximations that preserve thermodynamic consistency and stability.

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

The paper formulates hereditary constitutive models in linear viscoelasticity through an operator-theoretic lens. It proves that the history operator mapping strain histories to stresses is compact under natural assumptions on the relaxation kernel. Compactness directly implies the existence of optimal low-rank approximations quantified by Kolmogorov N-widths. These approximations correspond to internal-variable models that automatically satisfy the same thermodynamic consistency and stability as the original hereditary law while supplying explicit error bounds. The result supplies a rigorous basis for systematic reduced-order modeling in computational mechanics.

Core claim

The central claim is that the history operator is compact when the relaxation kernel satisfies natural assumptions. Compactness guarantees optimal finite-rank approximations in the Kolmogorov sense, which can be realized as internal-variable theories. The resulting reduced models inherit thermodynamic consistency, stability, and provable approximation bounds from the full hereditary representation, while clarifying the precise structural relation between integral hereditary laws and differential internal-variable formulations.

What carries the argument

The history operator, the linear map from past strain histories to current stress via convolution with the relaxation kernel; compactness of this operator is the property that enables the optimal low-rank reductions.

If this is right

  • Optimal approximations converge at the best possible rate given by the Kolmogorov N-widths of the history operator.
  • Thermodynamic consistency and stability of the full model carry over directly to every finite-rank reduction.
  • Explicit, computable bounds on the approximation error are available for any chosen rank.
  • The framework rigorously justifies replacing hereditary integrals with internal-variable evolution equations in numerical simulations.
  • Numerical tests confirm that the observed convergence matches the predicted optimal rates with respect to rank and sampling.

Where Pith is reading between the lines

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

  • The same compactness argument could be tested on kernels arising in other memory-dependent systems, such as viscoelastic fluids or thermal materials with memory.
  • The optimal history variables identified here might serve as a basis for adaptive rank selection in large-scale finite-element codes.
  • Connections to approximation theory for other hereditary operators, including those in fractional calculus or delay equations, remain to be explored.
  • The structural clarification between hereditary and internal-variable forms may guide the construction of thermodynamically consistent reduced models in nonlinear settings.

Load-bearing premise

The relaxation kernel must obey conditions that render the history operator compact on the chosen function spaces.

What would settle it

A concrete relaxation kernel meeting the natural assumptions for which the Kolmogorov N-widths cannot be achieved by any sequence of finite-rank internal-variable models, or for which the reduced models lose stability.

Figures

Figures reproduced from arXiv: 2604.16072 by Ignacio Romero, Michael Ortiz.

Figure 1
Figure 1. Figure 1: Exact relaxation function and optimal rank-N approximations. whence (70) ψn(τ ) = (Sϕn)(τ ) = k κn r 2 T e λτ cos(κnτ ). Optimal finite-rank approximation. Recall that (71) (SN ξ)(τ ) = X N n=1 ψn(τ )(ξ, ϕn), is the optimal rank-N approximation of S. The convergence of the sequence (SN ) can be illustrated by testing it with the forward step function hρ(τ ) starting at ρ. The corresponding relaxation funct… view at source ↗
Figure 2
Figure 2. Figure 2: Standard linear solid, (95), C0 = 2, C1 = 1, λ = λ0 = 1. From top to bottom, rows show a basis function en(τ ) (left) and the corresponding viscoelastic response e p n(τ ) ≡ Sen(τ ) (right), with n = −2, . . . , 2 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Standard linear solid, (95), C0 = 2, C1 = 1, λ = λ0 = 1, m = 15, M = 2m+1 = 31. Right eigenfunctions ϕM,k (blue) and left eigenfunctions ψM,k (orange). From top to bottom, left to right, k = 1, . . . , 10 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Standard linear solid, (95), C0 = 2, C1 = 1, λ = λ0 = 1. Singular values sM,k, k = 1, 2, . . . , 6 of SM as a func￾tion of M, the dimension of the truncated history subspace. Horizontal lines depict exact analytical values, M = ∞. we consider the step strain history (98) ϵt(τ ) = ( 1, 0 ≤ τ ≤ T /2, 0, T /2 < τ ≤ T. The inelastic strain history is (99) ϵ p t (τ ) = ( C1 C0 λ [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 5
Figure 5. Figure 5: Standard linear solid, (95), C0 = 2, C1 = 1, λ = λ0 = 1. In orange, inelastic strain (left) and weighted inelastic strain (right) when the model is loaded with a unit step strain at τ /T = 0.5. From top to bottom, solutions obtained with optimal history representations of dimension N = 1, 2, 4, 8, 16 and basis of dimension M = 2N + 1. The exact solution is shown in blue [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 6
Figure 6. Figure 6: Standard linear solid, (95), C0 = 2, C1 = 1, λ = λ0 = 1. Error in the inelastic strain measured in the H-norm as a function of N, the size of the eigen-basis, with a representation space of dimension M = 2N + 1. Opti￾mal vs. suboptimal Fourier-type representation. A reference curve with slope −0.28 is shown in green. produced by a step strain history is (101) ϵ p t (τ ) =    C1 C0λ  1 − e −λ(T /2−τ)  … view at source ↗
Figure 7
Figure 7. Figure 7: Representative volume element and its finite el￾ement mesh. The RVE has 43 “grains”, each of them with a different viscoelastic behavior and assigned to a different color. The finite element mesh employs 8 hexahedral ele￾ments per grain. assumptions, the relaxation modulus of grain k is (106) Rk(τ ) = µ∞ + X W i=1 µki e −τ /τki , µki = ηki τki , k = 1, . . . , Ng. where ηki and τki are the viscosities and … view at source ↗
Figure 8
Figure 8. Figure 8: RVE example. Histograms with the statistical distribution of the viscosity ηik and characteristic times τik of all the materials in the RVE. The Gamma distributions where the two random variables are sampled from are superposed on top of the data [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: RVE example. Imposed strains ϵxy(t) = ej (T −t) (left) vs. normalized mean stress τj (t)/µ∞ (right), −2 ≤ j ≤ 2 computed with the FE model [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: RVE example. Right and left eigenfunctions of the operator SM. From left to right, top to bottom, eigen￾functions ϕk, ψk with k = 1, . . . , 8, and M = 41 [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: RVE example. Parabolic shear strain imposed on the RVE and the average stress computed on the RVE [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: RVE example. Viscoelastic response of the RVE subject to shear strain (107). Left: inelastic strain obtained with the finite element model and with the optimal repre￾sentation using N eigenfunctions and M = 2N + 1. Right: H−norm of the error in the recovered inelastic strain [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: RVE example. Shear strain proportional to the step (98) and the average stress computed on the RVE [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: RVE example. Viscoelastic response of the RVE subject to shear strain (98). Left: inelastic strain obtained with the finite element model and with the optimal repre￾sentation using N eigenfunctions and M = 2N + 1. Right: H−norm of the error in the recovered inelastic strain [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
read the original abstract

We develop an operator-theoretic formulation of hereditary constitutive models and characterize optimal finite-rank internal-variable approximations in the sense of Kolmogorov $N$-widths. The history operator is shown to be compact under natural assumptions on the relaxation kernel, thereby admitting optimal low-rank approximations. The resulting reduced models inherit thermodynamic consistency, stability, and provable approximation bounds. An analysis clarifies the structural relation between hereditary representations and internal-variable theories and provides a rigorous basis for reduced-order modeling in computational mechanics. Selected numerical examples showcase optimal convergence of approximations with respect to rank and sampling.

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 / 1 minor

Summary. The manuscript develops an operator-theoretic formulation of hereditary constitutive models in linear viscoelasticity. It shows that the history operator is compact under natural assumptions on the relaxation kernel, which allows characterization of optimal finite-rank internal-variable approximations in the sense of Kolmogorov N-widths. The reduced models are claimed to inherit thermodynamic consistency, stability, and provable approximation bounds. The work also clarifies the structural relation between hereditary representations and internal-variable theories, and includes numerical examples demonstrating optimal convergence with respect to rank and sampling.

Significance. If the compactness result can be established with explicit kernel conditions, the paper would provide a rigorous foundation for reduced-order modeling in computational mechanics. The use of N-width optimality and inheritance of physical properties like thermodynamic consistency offers a principled alternative to heuristic reductions, with potential for provable error bounds in simulations.

major comments (1)
  1. The central claim that the history operator is compact (thereby admitting optimal low-rank approximations with inherited thermodynamic consistency and stability) rests on unspecified 'natural assumptions' on the relaxation kernel. No explicit conditions are provided regarding the kernel's decay rate, integrability class, monotonicity, or positive-definiteness in the relevant history space (e.g., to ensure the Volterra operator is compact). Without these, it is not possible to verify that the N-widths are attained by the proposed internal variables or that the reduced models remain thermodynamically consistent and stable.
minor comments (1)
  1. The numerical examples would benefit from explicit statements of the relaxation kernels employed, the precise error norms used to measure convergence, and the sampling strategy for the history variable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment, as well as for the constructive major comment. We address the concern point by point below and will revise the manuscript to incorporate explicit conditions.

read point-by-point responses

  1. Referee: The central claim that the history operator is compact (thereby admitting optimal low-rank approximations with inherited thermodynamic consistency and stability) rests on unspecified 'natural assumptions' on the relaxation kernel. No explicit conditions are provided regarding the kernel's decay rate, integrability class, monotonicity, or positive-definiteness in the relevant history space (e.g., to ensure the Volterra operator is compact). Without these, it is not possible to verify that the N-widths are attained by the proposed internal variables or that the reduced models remain thermodynamically consistent and stable.

    Authors: We thank the referee for this observation. The manuscript invokes 'natural assumptions' on the relaxation kernel in Section 2.1 to establish compactness of the history operator, but these are not enumerated as a self-contained list of mathematical conditions. To resolve the issue, we will add an explicit statement of the assumptions in a new subsection (Section 2.3): the kernel G(t) belongs to C^1([0,∞); L^∞(Ω)), is monotonically non-increasing, positive semi-definite (i.e., the associated quadratic form satisfies ∫ G(t-s)ε(s)·ε(t) ds ≥ 0), and satisfies ∫_0^∞ |G'(t)| dt < ∞ together with a decay condition ensuring the Volterra operator maps the history space into a compact subset (via Arzelà-Ascoli). Under these conditions, compactness follows from standard results on integral operators with weakly singular kernels. The proofs that the Kolmogorov N-widths are attained by the finite-rank internal-variable models (Theorem 3.4) and that thermodynamic consistency and stability are inherited (Theorems 4.1–4.2) rely only on these properties; we will add cross-references and a short verification paragraph. This revision will make the claims verifiable without altering the overall results. revision: yes

Circularity Check

0 steps flagged

No circularity: operator compactness and N-width optimality derived independently from kernel assumptions

full rationale

The paper's central derivation establishes compactness of the history operator under stated natural assumptions on the relaxation kernel, then invokes standard Kolmogorov N-width theory to obtain optimal low-rank approximations. These steps rely on functional-analytic results external to the paper and do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. Thermodynamic consistency and stability are inherited from the original hereditary model via the approximation construction, without circular redefinition. The derivation is self-contained against external operator theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the compactness of the history operator, which is asserted under unspecified natural assumptions on the relaxation kernel; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The history operator is compact under natural assumptions on the relaxation kernel
    Invoked to guarantee existence of optimal low-rank approximations

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