Stress distribution in elastic disks with a hole under uniaxial compression

Ken Okamura; Satoshi Takada; Yosuke Sato

REVIEW 3 major objections 5 minor 25 references

Stress distribution in elastic disks with a hole under uniaxial compression

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.21984 v1 pith:7OSIJKIH submitted 2025-05-28 cond-mat.soft physics.class-ph

Ken Okamura , Yosuke Sato , Satoshi Takada This is my paper

classification cond-mat.softphysics.class-ph MSC 74B0574G05

keywords hollowdiskstressdistributiondiametricloadingBraziliantestringelastodynamicsLaplacetransformplane

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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 $m=2,4,\dots$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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 $\sigma_0$.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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].

This 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.

Reading 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section 4, Eq. (14) with Eqs. (2) and (13)] 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 3, Eqs. (10) and (12)] 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 4, experimental comparison] 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.

minor comments (5)

[Section 3, Eq. (10)] 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 2, Eq. (2)] 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 4, Fig. 2] 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 4, Fig. 3] 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).
[General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (12) is an application of the authors' prior parameter-free analytical solution to a new Fourier loading profile.

full rationale

The derivation chain is not circular. The paper computes the static response of a hollow disk under a distributed diametral load by writing the load as the Fourier series in Eq. (2), inheriting the coefficient functions F_m,i, G_m,i, D_m, and N_m^(i) from the authors' earlier concentrated-load solution (Ref. [22]), and then forming Eq. (12) as a linear superposition weighted by the new coefficients ~ς_m of Eq. (14). The inherited coefficient functions are not fitted to data, not normalized against the target result, and not defined in terms of the claimed output; they solve the same linear elastodynamic boundary-value problem for a different boundary load. The new step is therefore a change of loading coefficients in an independently derived analytical framework, not a definitional or self-citational reduction. No parameters are fitted to experimental or numerical data, and the qualitative comparison with Ref. [24] does not determine any input. A separate internal correctness concern exists: Eq. (14) gives ~ς_m = 4θ0/π sinc(mθ0), which is twice the value obtained by expanding Eq. (13) in the form of Eq. (2) (the constant term would be 2σ0θ0/π, not 4σ0θ0/π), so the plotted stress magnitudes may be off by a factor of 2; however, this is a mathematical consistency error, not a circularity, because the claimed result does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central formulas depend on the authors' previous paper for the coefficient functions; that is the main unstated load. All other assumptions are standard mathematical tools for linear elastodynamics. No new physical entities are introduced and no parameters are fitted to data.

assumptions (8)

standard math Navier-Cauchy equations for linear isotropic plane-stress elasticity govern the deformation
Eq. (3) is the starting point; standard continuum mechanics.
standard math Helmholtz decomposition represents the displacement field as gradients and curls of scalar and vector potentials
Eq. (6); standard vector calculus theorem.
standard math Laplace transform and final value theorem give the static (τ→∞) solution
Eqs. (8) and (11), with l'Hopital's rule as stated; requires poles with negative real part for convergence.
standard math Modified Bessel functions Im and Km form a complete basis for the transformed wave equations in annular geometry
Eq. (9); standard solution of the modified Helmholtz equation with angular periodicity.
domain assumption The applied load has period π and even symmetry, so only even cosine modes appear
Eq. (2) and Eq. (13); appropriate for diametric loading with two symmetric patches.
domain assumption Traction boundary conditions Eq. (1) alone determine the stress field, with the m=1 rigid-translation mode silently set to zero
The paper sets m=1 coefficients to 0 (Eq. 10) without discussing the non-uniqueness of displacement for traction-only problems.
domain assumption Plane stress condition applies
Statement after Eq. (2): 'Assuming the plane stress condition'.
ad hoc to paper The coefficient functions F_m,i, G_m,i, D_m, and N_m(i) from Ref [22] are correct and remain valid for the distributed-load boundary condition
Eqs. (10) and (12) state they are 'the same as those used in Ref [22]' without restating or re-deriving them.

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Cite this review

Pith. "Pith review of Stress distribution in elastic disks with a hole under uniaxial compression." pith.science (2026). https://pith.science/paper/7OSIJKIH

@misc{pith2026250521984,
  author       = {Pith},
  title        = {Pith review of: Stress distribution in elastic disks with a hole under uniaxial compression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7OSIJKIH}},
  note         = {Machine review of arXiv:2505.21984}
}

read the original abstract

This paper investigates the stress and displacement distribution in a two-dimensional elastic hollow disk subjected to distributed diametric loading, extending our previous analysis of concentrated loading [Okamura et al. Strength Mater. 57, 102-114 (2025)]. The study provides deeper insights into the mechanical behavior of materials such as concrete and rock by examining the effects of load distribution on stress localization and displacement patterns. Using elastodynamic theory, we derive the static stress distributions and identify key differences from the concentrated loading case, particularly in the locations and magnitudes of stress extrema. This work contributes to a more comprehensive understanding of stress behavior in elastic disks under realistic loading conditions.

Figures

Figures reproduced from arXiv: 2505.21984 by the authors.

**Figure 1.** Schematic of our system. The stress acts on the outer surface of a hollow disk whose outer and inner radii are given by Ro and Ri , respectively. vector u satisfies the Navier-Cauchy equations [20]: ̺0 ∂ 2 ∂t 2 u = G∇ 2 u + G 1 + ν 1 − ν ∇ (∇ · u), (3) where G is the shear modulus, ν is Poisson’s ratio, and ̺0 is the mass density. Next, we introduce the longitudinal and transverse velocities asvL ≡ [2G/[(1 − ν)̺0] 1… view at source ↗

**Figure 3.** Plots of the (dimensionless) tensile stress along the loading line against ρ for (top) various inner radii ρi with θ0 = 0.25 and (bottom) various angles θ0 with ρi = 0.3, where Poisson’s ratio is fixed as ν = 0.3. Now, let us check the profile of the tensile on the loading axis for various ρi and θ0 in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [22]

Okamura, Y

K. Okamura, Y . Sato, and S. Takada, Strength Mater. 57, 102–114 (2025)

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Y .V . Tokovyy, K.-M. Hung, and C.-C. Ma, J. Math. Sci. 165, 342–354 (2010)

work page 2010

[3] [1]

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D.W . Hobbs, Int. J. Rock. Mech. Min. Sci. Geomech. Abstr. 1, 385–388 (1964)

work page 1964

[4] [2]

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J. Claesson and B. Bohloli, Int. J. Rock Mech. Min. Sci. 39, 8 (2002)

work page 2002

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J.C. Jaeger and N.G.W . Cook, Fundamentals of Rock Mechanics, Chapman and Hall (1979)

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S.P . Timoshenko and J.N. Goodier, Theory of Elas- ticity 3rd ed., McGraw-Hill (1970)

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K. Ramesh and K. Shins, Granul. Matter 24, 49 (2022)

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