{"id":"2c296bf5-4b30-4ddd-a483-1917bc11d701","arxiv_id":"2606.13297","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Eight 3D Eringen non-local models are derived from the 1D case via non-locality vector and nabla products, with compatibility conditions, Fourier-derived kernels, and isotropy classifications provided.","lead":"The paper extends a one-dimensional Eringen stress-gradient model to three dimensions by replacing a scalar non-locality parameter with a vector and using different vector products with the nabla operator, yielding eight models split into scalar and tensor types. These formulations could support more accurate stress predictions in microstructured materials where effects propagate non-locally in 3D.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the step that would matter for physical use, but that step lies outside the paper's formal claim. The derivations themselves contain no evident gap that would falsify the classification, compatibility conditions, or kernels.","tokens_in":1812,"tokens_out":286,"duration_ms":28721,"concrete_test":"Verify that each of the eight Fourier multipliers listed in the paper is exactly the symbol of the corresponding differential operator obtained from the vector products (i.e., replace ∇ by ik in the operator and confirm algebraic identity with the given multiplier).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a direct mathematical construction: eight differential operators are obtained by replacing the 1D second derivative with the eight possible scalar and tensor products formed from a fixed non-locality vector n and the nabla operator, the resulting operators are classified, compatibility conditions for stress symmetry are stated for the tensor cases, and the corresponding Fourier-space kernels are written down. Because the work is limited to this algebraic and transform-level construction, and because the paper explicitly derives the symmetry conditions and the kernels rather than asserting them, there is no internal inconsistency or hidden assumption whose failure would invalidate the listed results. Physical interpretability and reduction to known 3D Eringen models are outside the stated claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1938,"tokens_out":475,"duration_ms":30954,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§3"},{"comment":"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.","section":"§5"},{"comment":"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.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[],"tokens_in":1368,"tokens_out":56,"duration_ms":8837,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is straightforward: start from the 1D Eringen stress-gradient relation, replace the scalar length with a fixed vector n, and form the eight possible scalar or tensor products with the gradient operator. This yields three scalar-type and five tensor-type models. The paper then states the symmetry conditions required for the Cauchy stress in the tensor cases, writes down the Fourier-space kernels for each, and classifies the resulting isotropy (local and non-local) depending on the kernel.\n\nThe algebra is explicit and the Fourier step is standard, so the kernels and the isotropy distinctions follow directly. The observation that seven of the eight models retain both local and non-local stress terms is also useful to note. No fitted parameters appear, and the construction stays inside the Eringen framework without circular definitions.\n\nThe limitation is that the work remains a formal menu of operators. It does not supply physical motivation for choosing one product over another, does not compare predictions against existing 3D non-local models, and offers no route to experimental calibration. The derivations look free of algebraic contradictions on the given evidence, but the paper stops at the level of operator definition.\n\nSpecialists already using Eringen-type models in 3D might find the classification and kernels convenient. A general reader in continuum mechanics will not gain new physics or new predictive capability. The paper is coherent on its own terms and the math is reproducible from the stated steps.\n\nRecommendation: send it to peer review so the algebra and kernel derivations can be checked; it is not strong enough for a high-impact venue on its own.","headline":"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.","tokens_in":2435,"tokens_out":409,"would_cite":false,"duration_ms":16100,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"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.","keywords":["Eringen non-local model","stress-gradient","three-dimensional non-local elasticity","non-locality kernels","isotropy properties","compatibility conditions","non-local medium","Fourier transform"],"falsifier":"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.","tokens_in":2711,"feed_emoji":"","tokens_out":967,"duration_ms":38844,"temperature":0.7,"pith_summary":"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.","feed_headline":"Eight 3D Eringen non-local stress models formulated","feed_subtitle":"Non-locality vector and nabla products create scalar and tensor types, with kernels and isotropy derived for each.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Eight 3D Eringen models: three scalar five tensor types","Non-locality vector nabla products yield eight Eringen models","Three scalar and five tensor Eringen non-local models derived","Fourier kernels and isotropy for eight 3D Eringen models"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Eight 3D Eringen models: three scalar five tensor types","Non-locality vector nabla products yield eight Eringen models","Three scalar and five tensor Eringen non-local models derived","Fourier kernels and isotropy for eight 3D Eringen models"]},"model":"grok-4.3","cost_usd":0.004921,"raw_usage":{"total_tokens":2460,"prompt_tokens":768,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":49212000,"prompt_tokens_details":{"text_tokens":768,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1621,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":768,"tokens_out":71,"duration_ms":18995,"temperature":1.0,"reasoning_tokens":1621,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T04:49:24.992341+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}