{"id":"09f8e1c4-8715-47aa-b189-d02558aa180e","arxiv_id":"2505.21984","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The tensile hoop stress along the loading axis of a hollow disk under distributed diametral compression is not constant; its peaks and valleys move with the hole size and the width of the loaded arc.","lead":"This paper derives closed-form formulas for the stress and displacement fields in a hollow elastic disk squeezed by pressure over two finite arcs, a ring version of the Brazilian test, extending the authors' earlier point-load solution. It matters for interpreting tensile-strength tests on concrete and rock, where the load is applied through flat platens rather than a knife edge.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"A factor-of-2 normalization error in the Fourier coefficients of the patch load makes the quantitative stress magnitudes in Eq. (12)/Fig. 3 twice the value implied by the stated boundary condition.","rationale":"The central claim requires Eq. (12) to solve the stated boundary-value problem. The most direct check is to feed the stated boundary condition into Eq. (2) and compare with Eq. (14); this exposes a factor 2 that is independent of the correctness of the inherited coefficients. The error does not destroy the paper's qualitative thesis: because the solution is homogeneous of degree one in the Fourier coefficients, halving all ςm halves σ and u everywhere while leaving zero-crossings and extrema positions fixed. Thus the observation that the tensile stress varies along the loading line for a hollow disk, and that extrema positions shift with ρi and θ0, is likely robust. What fails is the quantitative claim: the stress magnitudes in Fig. 3, and the normalization of experimental comparison, are tied to a boundary load of amplitude 2σ0 rather than σ0. A conditional verdict is appropriate because the defect is a stated-coefficient inconsistency that can be repaired by a factor 1/2 and verified by an independent static solver; it does not require rejecting the method. This is why I would keep the reader's CONDITIONAL verdict rather than moving to ACCEPT or REJECT. The paper's self-containedness concern about Ref. [22] should also be addressed, but the normalization test should come first, since a numerical comparison cannot distinguish 'correct formula, wrong coefficient' from 'wrong formula' unless the coefficient definitions are restored.","tokens_in":6135,"tokens_out":7892,"duration_ms":84072,"concrete_test":"Directly integrate the Fourier series of Eq. (13) and compare with Eq. (14): if the correct coefficients are 2θ0/π sinc(mθ0), re-plot Fig. 3 with all ~ςm halved. To settle it independently, solve the static plane-stress problem numerically (e.g., finite element) for a hollow disk with ν=0.3, ρi=0.3, θ0=0.25, actual patch pressure σ0, and compare σθθ along the loading axis with Eq. (12) using both Eq. (14) and the corrected coefficients. The published curves should exceed the numerical curve by roughly a factor of 2, while the corrected curves should match; if instead the published curves match, Eq. (2) or Eq. (13) contains a compensating typo and the normalization statement should be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is internal: the applied load used in the plotted solution is not the load stated in Eq. (13). Expanding the box load σext(θ)=σ0 for |θ|≤θ0 (period π) in the form of Eq. (2), the constant term is (1/π)∫σext dθ = 2σ0θ0/π, and the cosine coefficients are 4σ0 sin(mθ0)/(π m). Matching Eq. (2) therefore requires ~ς0 = 2θ0/π and ~ςm = 2θ0/π sinc(mθ0). Eq. (14) gives ~ςm = 4θ0/π sinc(mθ0), exactly twice the correct value. Hence the boundary condition actually solved by Eq. (12) is a patch load of amplitude 2σ0, or, if one keeps σ0 as the nominal amplitude, every dimensionless stress and displacement in Figs. 2 and 3 is a factor of 2 too large. Because the solution is linear in the ~ςm, this rescales all curves but leaves the ρ-locations of the extrema unchanged; thus the qualitative claim that the tensile stress is non-constant survives, but the quantitative magnitudes in Fig. 3—explicitly part of the central assertion—are invalid. The paper's reliance on Ref. [22] for Dm and Nm(i) is a secondary concern; the factor-of-2 check should be performed after restoring those definitions to rule out an additional normalization mismatch from the inherited coefficients.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript extends the authors' earlier elastodynamic solution for concentrated diametral loading of a two-dimensional elastic hollow disk to a distributed (patch) load. It states the boundary-value problem, expands the patch load in a Fourier series, imports the coefficient functions from Ref. [22], and gives static displacement and stress formulas via the final-value theorem. The paper then plots the displacement magnitude and principal-stress difference, and studies the hoop stress on the loading axis as functions of inner radius and load half-width, concluding that this stress is not constant across the cross-section, unlike the solid-disk case. A qualitative comparison with a published photoelastic experiment is also reported.","tokens_in":6419,"tokens_out":6188,"duration_ms":65472,"significance":"If the derivation were correct, the paper would provide a compact closed-form solution for the hollow-disk (ring-type) Brazilian test with distributed loading, with explicit predictions for how the positions and magnitudes of stress extrema depend on the hole radius and the load-distribution width. The parameter sweeps in Fig. 3 are a useful addition to the earlier concentrated-load analysis. Since the work is entirely analytic, its value depends on the formulas being self-contained and quantitatively correct. At present, the central quantitative assertion is compromised by a factor-of-2 normalization inconsistency, and the core coefficient functions are imported without definition, so the manuscript is not yet trustworthy in its current form.","major_comments":[{"comment":"The coefficients ~ς_m = 4 θ0/π sinc(mθ0) do not match the Fourier convention in Eq. (2) for the box load in Eq. (13). For a period-π box of amplitude σ0, the required coefficients are ~ς_0 = 2 θ0/π and ~ς_m = (2 θ0/π) sinc(mθ0), which are exactly half of Eq. (14). Inserting Eq. (14) into Eq. (2) gives a constant term 4 σ0 θ0/π instead of 2 σ0 θ0/π, so the boundary condition actually solved by Eq. (12) is a patch load of amplitude 2σ0 rather than σ0. Since the solution is linear in the ~ς_m, all dimensionless displacements and stresses in Figs. 2 and 3 are a factor of 2 too large relative to the stated load amplitude. The locations of the extrema are unaffected by this rescaling, so the qualitative claim that the tensile stress is non-constant survives, but the quantitative magnitudes in Fig. 3, which are part of the paper's central assertion, are invalid and need correction and re-plotting.","section":"Section 4, Eq. (14) with Eqs. (2) and (13)"},{"comment":"The functions F_{m,i}, G_{m,i}, D_m, and N_m^{(i)} are not defined in this manuscript; the text only states that they are 'the same as those used in Ref. [22]'. These functions encode the entire boundary-value problem, including the transfer from the concentrated-load case to the distributed-load case, so the central equations are not self-contained. A reader cannot verify the derivation without the prior paper, and the claim that the same coefficient functions remain valid for the new boundary conditions is not independently checked. Please reproduce the definitions in an appendix or provide explicit expressions for the static limits used in Eq. (12), so that the solution can be evaluated and checked directly.","section":"Section 3, Eqs. (10) and (12)"},{"comment":"The comparison with Ref. [24] is purely qualitative and the paper itself notes that no numerical data are available for a direct quantitative check. Given the factor-of-2 normalization error, the claimed consistency with the experimental patterns does not constrain the quantitative amplitudes. After correcting the normalization, I strongly recommend adding an independent quantitative benchmark, for example against a finite-element solution of the same boundary-value problem or against tabulated experimental stress data, before the magnitudes in Figs. 2 and 3 are presented as validated results.","section":"Section 4, experimental comparison"}],"minor_comments":[{"comment":"The notation is inconsistent: the text refers to F_{m,i}, G_{m,i} with i = 0,...,3, but the displayed formulas use only F_{0,1}, G_{0,1}, N_{m,1}, N_{m,2}, N_{m,3}, N_{m,4}; please clarify the meaning of the index i and align the notation.","section":"Section 3, Eq. (10)"},{"comment":"The Fourier convention with the explicit factor 2σ0 and the ~ς_0/2 term is nonstandard and easy to misread; a short derivation of the box-load coefficients would remove ambiguity, especially after the normalization error is fixed.","section":"Section 2, Eq. (2)"},{"comment":"The color maps have no colorbar or numeric scale, even though the paper emphasizes quantitative stress magnitudes; adding colorbars with dimensionless values would make the figures interpretable.","section":"Section 4, Fig. 2"},{"comment":"The quantity plotted is described only as 'the tensile stress'; the caption should state explicitly that it is the dimensionless hoop stress ~σθθ along the loading axis, and the values of θ0 should be labeled with their units (radians).","section":"Section 4, Fig. 3"},{"comment":"There are several typographical and spacing artifacts in the text (for example, 'uniax ial' in the title page, and '-' etc.); a careful proofreading pass is needed.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends very heavily on the authors' own Ref. [22] for all non-trivial coefficient functions, and the present text reads as an incremental parameter change rather than a fully self-contained derivation. The factor-of-2 normalization error is a clear internal inconsistency that must be fixed, and the missing coefficient definitions must be supplied before the paper can be assessed on its merits. The qualitative finding is plausible, but the quantitative content is not yet reliable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is modest but real: the same elastodynamic machinery used in the authors' earlier concentrated-load paper is applied to a finite-width diametral load on a hollow elastic disk. The governing equations and the coefficient functions are inherited from Ref. [22], and the new content is essentially the Fourier coefficients of a rectangular patch plus a parameter scan over hole radius and load width. That is a legitimate incremental step, and the qualitative claim—that tensile stress along the loading axis is not constant, with extrema shifting as ρi and θ0 change—is consistent with what the linear-elastic boundary-value problem should give. The paper is honest about its lineage and does not oversell the comparison with Ref. [24], which is only qualitative.\n\nThe load-bearing problem is internal. Eq. (13) states the applied compression has amplitude σ0 over |θ|≤θ0, but Eq. (14) gives ~ςm = 4θ0/π sinc(mθ0) everywhere. Expanding the box load in the form of Eq. (2) gives ~ς0 = 2θ0/π and ~ςm = 2θ0/π sinc(mθ0). So every ~ςm is twice what it should be. Because the displacement and stress formulas are linear in ~ςm, that means every curve in Figs. 2 and 3—and every reported magnitude—corresponds to a patch load of amplitude 2σ0, not σ0. The locations of the extrema survive because they are scale-invariant, so the qualitative picture in the paper is probably right. But as written, the quantitative values are a factor of 2 too large, and this is precisely the kind of error a referee should catch. A second, lesser issue is that Dm, Nm(i), and the other coefficient functions are not defined in this paper; the central equations are not self-contained. I would not call that fatal for a sequel, but it does make checking the normalization harder, and it means the paper can't stand alone without Ref. [22].\n\nThis is a short, useful contribution for people working on Brazilian and ring tests, but it needs a concrete fix before it can be trusted. I would send it to peer review—a competent referee will spot the factor of two—and the authors should also provide the missing coefficient definitions or an independent numerical check (finite-element would do) to verify magnitudes. With those changes, it would be a fine specialist note.\n\nReading group? Maybe, as a clean example of a normalization slip surviving into a plot. I probably wouldn't cite it myself until the factor-of-two is resolved.","headline":"A clean analytical extension of the authors' prior point-load solution to a finite-width patch, but Eq. (14) carries a factor-of-2 normalization error that invalidates all plotted stress magnitudes until corrected.","tokens_in":6921,"tokens_out":1865,"would_cite":false,"duration_ms":20154,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74B05","74G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A hollow elastic disk under distributed diametral loading has a closed-form static stress solution, and the tensile stress along the loading axis varies with position instead of staying constant.","keywords":["hollow disk","stress distribution","diametric loading","Brazilian test","ring test","elastodynamics","Laplace transform","plane stress"],"falsifier":"Compute the constant term of the Fourier series in Eq. (2) with coefficients (14): for the patch (13) it gives $2\\sigma_0\\theta_0/\\pi$ rather than $\\sigma_0\\theta_0/\\pi$, so an independent numerical solution (e.g., finite elements) of the stated boundary-value problem for a chosen $\\rho_i$, $\\theta_0$, and $\\nu$ would show whether the plotted stress magnitudes in Fig. 3 are off by this normalization factor or correct.","tokens_in":5895,"feed_emoji":"⭕","tokens_out":9511,"duration_ms":82177,"temperature":0.7,"pith_summary":"This paper extends an earlier closed-form elastodynamic analysis of a hollow elastic disk under a concentrated diametric force to the case of two diametrically opposite patches of constant pressure $\\sigma_0$. Working in plane stress, the authors solve the Navier-Cauchy equations through Helmholtz potentials and Laplace transforms, then take the long-time limit to obtain the static displacement and stress fields. Their central claim is that these closed-form expressions, Eqs. (12) with load coefficients (14), are the true static solution for the patch-loaded hollow disk, and that the tensile stress along the loading axis is not constant there, in contrast to the solid disk. If correct, the results give concrete predictions for where tensile and compressive stress extrema occur in ring-test specimens as functions of hole size and load width, which matters for interpreting Brazilian-type tensile tests on concrete and rock.","feed_headline":"Hollow-disk tensile stress along the load line is not constant","feed_subtitle":"New closed-form solution shows where stress peaks sit in ring-test specimens as hole size and load width vary.","key_machinery":"The central machinery is the Laplace-transformed elastodynamic representation of the displacement in a hollow disk: the displacement is decomposed as $\\mathbf{u}=\\nabla\\varphi+\\nabla\\times\\mathbf{A}$, the scalar and vector potentials satisfy wave equations whose Laplace-transformed general solutions are written as sums of modified Bessel functions $I_m(\\rho s)$ and $K_m(\\rho s)$, and the unknown coefficients are fixed by the stress boundary conditions at the outer and inner surfaces. The static solution is then extracted by the final value theorem, taking the $s\\to 0$ limit of the Laplace-transformed fields, which converts the Bessel-function expressions into the algebraic coefficients $D_m$ and $N_m^{(i)}$ that appear in Eq. (12). For the distributed load, the angular dependence of the patch is encoded in the Fourier coefficients $\\tilde{\\varsigma}_m = 4\\theta_0\\operatorname{sinc}(m\\theta_0)/\\pi$, so the entire answer is an infinite series over even harmonics $m=2,4,\\dots$.","core_discovery":"The paper claims that for a two-dimensional elastic hollow disk with outer radius $R_o$ and inner radius $R_i$, loaded by a constant radial compression $\\sigma_0$ over two arcs $|\\theta|\\le\\theta_0$ and $|\\theta-\\pi|\\le\\theta_0$, the dimensionless static stress and displacement fields are given exactly by Eq. (12), with Fourier coefficients $\\tilde{\\varsigma}_m = 4\\theta_0\\operatorname{sinc}(m\\theta_0)/\\pi$ for even $m$ and zero for odd $m$. The derivation starts from linear elastodynamics, uses Helmholtz decomposition and Laplace transforms to obtain the dynamic potentials, and then evaluates the $s\\to 0$ limit via the final value theorem to reach the static solution. The central observation is that the resulting tensile stress $\\tilde{\\sigma}_{\\theta\\theta}$ along the loading axis varies with radius and depends on both the inner-radius ratio $\\rho_i$ and the load half-width $\\theta_0$; this is explicitly contrasted with the solid disk, where the tensile stress on that line is constant. The solution also reproduces the earlier concentrated-load result when the Fourier coefficients reduce to $\\tilde{\\varsigma}_m = 1/\\pi$, and the computed second stress difference shows spatial patterns consistent with previously reported photoelastic experiments.","pith_inferences":["If the Fourier coefficients in Eq. (14) are inserted into Eq. (2), the constant term is $2\\sigma_0\\theta_0/\\pi$ rather than $\\sigma_0\\theta_0/\\pi$ for the patch in Eq. (13), so the plotted magnitudes may carry a shape-independent normalization error; the qualitative conclusion about non-uniformity would not change.","Because tensile stress is position-dependent, standard ring-test strength formulas that read a single value may need to specify the radial coordinate; comparing two specimens with different hole sizes at the same nominal load would be a direct test.","The same Laplace-transform and potential machinery could be applied to other hole geometries, such as elliptical holes, or to asymmetric load patches by changing only the Fourier coefficients and boundary matching.","The paper does not provide numerical data from the experimental comparison, so a quantitative check would require digitizing the reported experimental displacement or stress fields and overlaying the analytical curves."],"forward_implications":["In ring-test and Brazilian-type tests on concrete and rock, the tensile stress along the loading axis cannot be treated as uniform; the location of the maximum must be read off the radius-dependent curve.","For a fixed hole size, increasing the load-patch width $\\theta_0$ shifts the tensile-stress profile and changes the extrema, so measured tensile strengths depend on the contact geometry of the loading platens.","As the patch width shrinks toward a point load, the distributed-load solution reduces to the earlier concentrated-load result, providing a consistency check for both.","The closed-form expressions allow photoelastic fringe patterns, proportional to the second stress difference, to be predicted for arbitrary hole sizes and load widths without numerical simulation.","Because the solution is obtained through the static limit of an elastodynamic solution, the same Fourier-Bessel representation is in principle available for extending the results to transient loading."],"supporting_citations":[{"why":"Supplies the coefficient functions $F_{m,i}$, $G_{m,i}$, $D_m$, and $N_m^{(i)}$ that the static solution Eq. (12) takes over without restatement.","marker":"[22]"},{"why":"Provides the solid-disk stress distribution against which the hollow-disk result is explicitly contrasted.","marker":"[4]"},{"why":"States the Navier-Cauchy equations used as the starting point of the derivation.","marker":"[20]"},{"why":"Gives the modified Bessel functions $I_m$ and $K_m$ used in the Laplace-transformed potential solutions.","marker":"[21]"},{"why":"Connects the second stress difference to photoelastic interference fringes used for the qualitative experimental comparison.","marker":"[23]"},{"why":"Reports the experimental stress features (sharp inner-boundary variation and horizontal stripe patterns) that the authors compare with their analytical results.","marker":"[24]"}],"fun_headline_variants":["Hollow disk: tensile stress on load axis varies with hole size","Elastic ring under compression: stress peaks shift with hole size","Constant stress on load line? Not for hollow disks","Ring test: tensile stress along load line varies with hole radius","Hollow-disk stress: load-axis tension depends on hole size and load width"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result assumes that the coefficient functions carried over from the earlier concentrated-load treatment remain valid for a distributed patch load and that the Fourier series in Eq. (2) with coefficients (14) has the same amplitude convention as the stated patch stress $\\sigma_0$.","fun_headline_variants_meta":{"raw":{"variants":["Hollow disk: tensile stress on load axis varies with hole size","Elastic ring under compression: stress peaks shift with hole size","Constant stress on load line? Not for hollow disks","Ring test: tensile stress along load line varies with hole radius","Hollow-disk stress: load-axis tension depends on hole size and load width"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00079,"raw_usage":{"total_tokens":3466,"prompt_tokens":911,"completion_tokens":2555,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":2466}},"tokens_in":527,"tokens_out":2555,"duration_ms":18908,"temperature":1.0,"reasoning_tokens":2466,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:19:00.315916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the constant term of the Fourier series in Eq. (2) with coefficients (14): for the patch (13) it gives $2\\sigma_0\\theta_0/\\pi$ rather than $\\sigma_0\\theta_0/\\pi$, so an independent numerical solution (e.g., finite elements) of the stated boundary-value problem for a chosen $\\rho_i$, $\\theta_0$, and $\\nu$ would show whether the plotted stress magnitudes in Fig. 3 are off by this normalization factor or correct.","supporting_citations":[{"cited_title":"Okamura, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the coefficient functions $F_{m,i}$, $G_{m,i}$, $D_m$, and $N_m^{(i)}$ that the static solution Eq. (12) takes over without restatement."},{"cited_title":"Timoshenko and J.N","cited_arxiv_id":null,"evidence_quote":"Provides the solid-disk stress distribution against which the hollow-disk result is explicitly contrasted."},{"cited_title":"Fung and P","cited_arxiv_id":null,"evidence_quote":"States the Navier-Cauchy equations used as the starting point of the derivation."},{"cited_title":"Abramowitz and I.A","cited_arxiv_id":null,"evidence_quote":"Gives the modified Bessel functions $I_m$ and $K_m$ used in the Laplace-transformed potential solutions."},{"cited_title":"A Trea- tise on Photo-Elasticity","cited_arxiv_id":null,"evidence_quote":"Connects the second stress difference to photoelastic interference fringes used for the qualitative experimental comparison."},{"cited_title":"Tokovyy, K.-M","cited_arxiv_id":null,"evidence_quote":"Reports the experimental stress features (sharp inner-boundary variation and horizontal stripe patterns) that the authors compare with their analytical results."}],"review_version":1}