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arxiv: 2606.22202 · v1 · pith:EBY3UX2Anew · submitted 2026-06-20 · ⚛️ physics.class-ph · cond-mat.mtrl-sci

On Potentials and Complementary Potentials in One-Dimensional Nonlocal Integral Formulations

Marko \v{C}ana{\dj}ija , Ante Skoblar This is my paper

Pith reviewed 2026-06-26 11:09 UTC · model grok-4.3

classification ⚛️ physics.class-ph cond-mat.mtrl-sci
keywords nonlocal integral formulationspotentialscomplementary potentialsLegendre transformationstrain-drivenstress-drivenone-dimensional
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The pith

Legendre transformation produces equivalent potentials and complementary potentials for one-dimensional nonlocal strain-driven and stress-driven formulations.

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

The paper shows how to construct complementary potentials for both the pure strain-driven and pure stress-driven nonlocal integral models in one dimension, using the Legendre transformation to establish their equivalence. This supplies the missing variational structure for the stress-driven case and confirms that the two formulations are interchangeable at the level of potentials. The same transformation also permits definition of a mixed stress-strain potential, although the authors demonstrate that this mixed version inherits the same ill-conditioning already known for the pure strain-driven version. An explicit example is solved to verify that the stress-driven model derived from the potential matches the one derived from the complementary potential.

Core claim

The equivalent formulations are obtained by resorting to the Legendre transformation, and their equivalence is proved. It is also shown that these results can be used to postulate a novel potential, i.e. a kind of mixed stress-strain potential, which is, however, as ill-conditioned as the pure strain-driven formulation. Finally, an example is given that practically confirms that the stress-driven formulations resulting from the potential and the complementary potential are equivalent.

What carries the argument

Legendre transformation applied to the nonlocal integral operators, which converts the known strain-driven potential into a complementary potential for the stress-driven formulation while preserving equivalence.

If this is right

  • Strain-driven and stress-driven nonlocal formulations become variationally equivalent once their potentials and complementary potentials are related by the Legendre transformation.
  • A mixed stress-strain potential can be defined directly from the transformed quantities.
  • The mixed potential remains ill-conditioned to the same degree as the pure strain-driven potential.
  • Numerical solutions of stress-driven problems obtained from either the potential or the complementary potential coincide.

Where Pith is reading between the lines

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

  • The equivalence may allow modelers to switch between strain-driven and stress-driven descriptions without changing the underlying energy functional.
  • Extension of the same Legendre construction to two or three dimensions would supply complementary potentials for nonlocal continuum models in higher dimensions.
  • The persistent ill-conditioning of the mixed potential indicates that any practical implementation would still require additional regularization or stabilization techniques.

Load-bearing premise

The nonlocal integral operators and associated potentials admit a Legendre transformation that yields well-defined complementary potentials and preserves equivalence.

What would settle it

A boundary-value problem in which the total energy computed from the potential differs from the energy computed from the complementary potential after the Legendre transformation is applied.

read the original abstract

The present research presents potentials and complementary potentials used in the one-dimensional nonlocal integral formulations. The pure stress and the pure strain nonlocal formulations were considered. While the potential used in the strain driven formulation is well known, the complementary potential has not yet been presented in the literature. The same applies to the stress driven formulation. The equivalent formulations are obtained by resorting to the Legendre transformation, and their equivalence is proved. It is also shown that these results can be used to postulate a novel potential, i.e. a kind of mixed stress-strain potential, which is, however, as ill-conditioned as the pure strain-driven formulation. Finally, an example is given that practically confirms that the stress-driven formulations resulting from the potential and the complementary potential are equivalent.

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 / 2 minor

Summary. The paper claims to introduce complementary potentials for the one-dimensional nonlocal integral strain-driven and stress-driven formulations (the strain-driven potential being already known), obtain equivalent formulations via the Legendre transformation with a proof of equivalence, postulate a novel mixed stress-strain potential (noted as ill-conditioned), and provide an example confirming that the stress-driven formulations derived from the potential and complementary potential are equivalent.

Significance. If the central derivations hold with the required regularity conditions made explicit and verified, the work would supply previously missing complementary potentials for these nonlocal integral models and demonstrate their equivalence, along with a mixed potential. This could strengthen the variational framework for nonlocal continuum mechanics in one dimension. The example provides concrete verification of the stress-driven case.

major comments (2)
  1. [Legendre transformation and equivalence proof (abstract and main derivations)] The central claim relies on the Legendre transformation yielding well-defined complementary potentials and preserving equivalence between strain-driven and stress-driven formulations. However, no conditions (e.g., convexity, coercivity, or positive-definiteness of the nonlocal kernels or energy functionals) are stated or verified to ensure the transform is bijective and the resulting potentials are equivalent; nonlocal integral operators can violate these in general, making the equivalence conditional rather than general. This is load-bearing for the proof and the mixed-potential construction.
  2. [Example section] The example is said to 'practically confirm' equivalence of the stress-driven formulations, but without quantitative metrics (e.g., error norms between solutions from potential vs. complementary potential, or explicit kernel definitions), it is unclear whether it tests the general case or only a special convex instance.
minor comments (2)
  1. [Abstract] The abstract states that 'the complementary potential has not yet been presented in the literature' for the strain-driven case and similarly for stress-driven; explicit citations to prior work on nonlocal potentials would clarify the novelty.
  2. [Introduction and formulation sections] Notation for the nonlocal integral operators and kernels should be introduced with explicit definitions early in the manuscript to aid readability of the potential expressions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. The points raised highlight areas where explicit assumptions and quantitative verification can strengthen the presentation. We will revise the manuscript accordingly and address each major comment below.

read point-by-point responses

  1. Referee: [Legendre transformation and equivalence proof (abstract and main derivations)] The central claim relies on the Legendre transformation yielding well-defined complementary potentials and preserving equivalence between strain-driven and stress-driven formulations. However, no conditions (e.g., convexity, coercivity, or positive-definiteness of the nonlocal kernels or energy functionals) are stated or verified to ensure the transform is bijective and the resulting potentials are equivalent; nonlocal integral operators can violate these in general, making the equivalence conditional rather than general. This is load-bearing for the proof and the mixed-potential construction.

    Authors: We agree that the regularity conditions required for the Legendre transform to be bijective should be stated explicitly. In the revised manuscript we will insert a new preliminary subsection (before the main derivations) that lists the standing assumptions: convexity and coercivity of the strain energy functional, positive-definiteness of the nonlocal kernel, and sufficient smoothness to guarantee the existence of the inverse operator. Under these hypotheses the equivalence proof already given in the paper holds; the revision will simply make the hypotheses visible rather than implicit. We do not claim the result is unconditional, and the added statement will clarify the scope. revision: yes

  2. Referee: [Example section] The example is said to 'practically confirm' equivalence of the stress-driven formulations, but without quantitative metrics (e.g., error norms between solutions from potential vs. complementary potential, or explicit kernel definitions), it is unclear whether it tests the general case or only a special convex instance.

    Authors: We accept that the example would be more convincing with quantitative measures. In the revision we will (i) state the explicit kernel (a standard exponential kernel with given length-scale parameter) and (ii) report L2-norm differences between the displacement fields obtained from the potential-based and complementary-potential-based stress-driven formulations. These norms will be shown to be on the order of machine precision, confirming numerical equivalence for the chosen convex instance that satisfies the assumptions listed in the new preliminary subsection. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external Legendre transform to known potentials

full rationale

The paper starts from the well-known strain-driven potential, applies the standard Legendre transformation (an external mathematical operation) to obtain complementary potentials for both strain-driven and stress-driven nonlocal integral formulations, and proves equivalence. No step reduces a claimed prediction or result to a self-definition, fitted input, or self-citation chain; the central claims rest on the properties of the Legendre transform applied to the given integral operators rather than on any quantity being defined circularly in terms of its output. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on the applicability of the Legendre transformation to the nonlocal potentials without introducing new free parameters or invented entities; the central claim rests on standard properties of convex duality.

axioms (1)
  • domain assumption The potentials arising in the pure strain and pure stress nonlocal integral formulations are such that the Legendre transformation yields equivalent complementary potentials.
    Invoked to obtain and prove equivalence of the formulations.

pith-pipeline@v0.9.1-grok · 5662 in / 1204 out tokens · 49158 ms · 2026-06-26T11:09:41.624248+00:00 · methodology

discussion (0)

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Reference graph

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