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The one-dimensional Eringen stress-gradient model extends to eight three-dimensional versions by replacing the scalar non-locality parameter with a vector and using different vector products with the nabla operator.

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2026-06-27 04:49 UTC pith:AILQ7TTL

load-bearing objection This paper gives a clean algebraic extension of the 1D Eringen model to eight 3D variants using a non-locality vector and different nabla products, with explicit Fourier kernels and isotropy checks.

arxiv 2606.13297 v1 pith:AILQ7TTL submitted 2026-06-11 physics.class-ph

Formulation of stress-gradient models describing three-dimensional non-local medium

Sla{\dj}an Jeli\'c , Du\v{s}an Zorica This is my paper
classification physics.class-ph
keywords Eringen non-local modelstress-gradientthree-dimensional non-local elasticitynon-locality kernelsisotropy propertiescompatibility conditionsnon-local mediumFourier transform
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper starts from the one-dimensional Eringen stress-gradient non-local model and generalizes it to three dimensions. It replaces the scalar non-locality parameter with a non-locality vector and the second-order derivative with various vector products involving the nabla operator. This produces eight models divided into three scalar-type and five tensor-type based on the resulting non-locality operator. The work derives compatibility conditions for symmetry of the stress tensor, non-locality kernels using Fourier transforms, and examines isotropy properties for each model. A sympathetic reader would care because these models provide ways to describe non-local effects in three-dimensional materials where local and non-local contributions to stress can be accounted for differently.

Core claim

Based on one-dimensional Eringen stress-gradient non-local model, by considering non-locality vector and nabla operator instead of non-locality scalar parameter and second order derivative, eight three-dimensional Eringen non-local models are formulated and classified into two groups: three scalar- and five tensor-type non-local models, according to the type of used non-locality operator which is obtained via various vector products of non-locality vector and nabla operator. The compatibility conditions ensuring symmetricity of Cauchy stress tensor in the case of the tensor-type model are derived. Furthermore, using the Fourier integral transform with respect to spatial coordinates, non-loca

What carries the argument

The non-locality operator obtained via various vector products of the non-locality vector and the nabla operator, which classifies models as scalar-type or tensor-type and governs the non-local contribution to the Cauchy stress tensor.

Load-bearing premise

The one-dimensional Eringen stress-gradient model can be directly extended to three dimensions by substituting a non-locality vector for the scalar parameter and replacing the second-order derivative with vector products involving the nabla operator, while preserving physical interpretability and symmetry requirements.

What would settle it

A direct check would be whether stress fields computed from the derived non-locality kernels match measured distributions in a three-dimensional specimen of a non-local elastic material under controlled loading.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Compatibility conditions are derived to ensure the Cauchy stress tensor remains symmetric for tensor-type models.
  • Non-locality kernels (Green's functions) are obtained via Fourier transform for each model.
  • All but one scalar-type model include both local and non-local contributions to the Cauchy stress tensor.
  • All scalar-type models are isotropic with one non-locally isotropic and two non-locally anisotropic; all tensor-type models are anisotropic with two non-locally isotropic and three non-locally anisotropic along specific directions.
  • Isotropy and non-local isotropy properties depend on the choice of non-locality kernel.

Where Pith is reading between the lines

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

  • The models could be applied to predict directional non-local effects in 3D structures such as composite panels or biological tissues.
  • Numerical implementation of the different operator types would allow comparison of predicted strain fields against finite-element simulations of heterogeneous materials.
  • The separation into models that do or do not prefer a non-locality direction suggests a route for designing materials with controlled anisotropy in non-local response.
  • Time-dependent or coupled-field versions could be obtained by applying the same vector-product construction to dynamic or thermoelastic equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript extends the one-dimensional Eringen stress-gradient model to three dimensions by replacing the scalar non-locality parameter with a non-locality vector n and the second-order derivative with eight possible scalar and tensor products formed from n and the nabla operator. This yields three scalar-type and five tensor-type models. Compatibility conditions ensuring symmetry of the Cauchy stress are derived for the tensor-type cases. Non-locality kernels (Green's functions) are obtained via Fourier integral transform for each model. Isotropy and non-local isotropy are analyzed: all scalar-type models are isotropic (one non-locally isotropic, two non-locally anisotropic), while all tensor-type models are anisotropic (two non-locally isotropic, three non-locally anisotropic along specific directions). All but one scalar-type model include both local and non-local stress contributions.

Significance. The work supplies an explicit algebraic classification and a set of closed-form Fourier kernels for a family of 3D stress-gradient operators generated systematically from vector products. The derivation of symmetry compatibility conditions and the isotropy classification constitute concrete, usable results for further analytical or numerical work in non-local continuum mechanics. The absence of fitted parameters and the direct construction from the cited 1D Eringen model are strengths that make the kernels immediately testable against known limits.

minor comments (3)
  1. [§3] A compact table listing the eight operators, their type (scalar/tensor), the explicit vector-product definition, and the resulting compatibility condition (where applicable) would improve cross-reference between the classification in §3 and the kernel derivations in §4.
  2. [§5] The statement that 'all tensor-type models are anisotropic' should be accompanied by a brief remark on whether the anisotropy is with respect to the material symmetry group or solely induced by the fixed direction of n; the current wording leaves this distinction implicit.
  3. Notation for the non-locality vector is introduced as n but occasionally appears as a bold vector without consistent font; a single definition in the notation section would eliminate ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed and positive summary of our work, the assessment of its significance, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Direct algebraic extension; no circularity

full rationale

The derivation begins from the cited one-dimensional Eringen model and performs an explicit algebraic substitution: replace the scalar non-locality parameter with a fixed vector n and the second-order derivative with the eight possible scalar/tensor products formed from n and ∇. Compatibility conditions for stress symmetry and the Fourier-space kernels are then derived directly from these operators. No parameter is fitted to data, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or self-definition. The construction is self-contained vector-calculus and transform work whose outputs are independent of any fitted values or prior author theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central construction rests on the assumption that the 1D Eringen model generalizes via vector substitution and that the resulting operators yield physically admissible stress tensors; no free parameters are fitted in the abstract, and no new entities are postulated beyond the non-locality vector already present in the 1D model.

axioms (2)
  • standard math Fourier integral transform exists and inverts for the spatial kernels
    Invoked to obtain non-locality kernels for each model
  • domain assumption Cauchy stress tensor must remain symmetric
    Used to derive compatibility conditions for tensor-type models

pith-pipeline@v0.9.1-grok · 5803 in / 1433 out tokens · 40668 ms · 2026-06-27T04:49:24.992341+00:00 · methodology

0 comments
read the original abstract

Based on one-dimensional Eringen stress-gradient non-local model, by considering non-locality vector and nabla operator instead of non-locality scalar parameter and second order derivative, eight three-dimensional Eringen non-local models are formulated and classified into two groups: three scalar- and five tensor-type non-local models, according to the type of used non-locality operator which is obtained via various vector products of non-locality vector and nabla operator. The compatibility conditions ensuring symmetricity of Cauchy stress tensor in the case of the tensor-type model are derived. Furthermore, using the Fourier integral transform with respect to spatial coordinates, non-locality kernels (Green's functions), reflecting non-locality character of the material, are derived for each of the proposed models. Except for the one scalar-type model, all other models account for both local and non-local contributions to Cauchy stress tensor. Additionally, the isotropy of proposed models, as well as their non-local isotropy properties, both depending on non-locality kernel, are examined. All scalar-type models are isotropic, such that one of them is non-locally isotropic and two of them correspond to a non-locally anisotropic body, while all tensor-type models are anisotropic, such that there are two models that do not prefer direction of non-locality, thus corresponding to a non-locally isotropic body, whereas three models correspond to a body exhibiting non-locality along a specific direction(s), thus corresponding to a non-locally anisotropic body.

discussion (0)

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Reference graph

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