Revisiting Stress Analysis in a Three-Dimensional Elastic Hollow Sphere under Uniaxial Compression via the Inverse Laplace Transform Expressions within an Elastodynamic Framework

Satoshi Takada; Shintaro Hokada

REVIEW 3 major objections 6 minor 17 references

This paper claims that the static displacement and stress fields of a three-dimensional elastic hollow sphere under uniaxial compression can be written explicitly as Legendre series, obtained as the long-time limit of an elastodynamic solut

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2026-08-03 09:57 UTC pith:AZZGUXZK

load-bearing objection A useful set of explicit static coefficients for the hollow sphere, but the claimed derivation via the final value theorem in an undamped elastodynamic problem is not valid as stated. the 3 major comments →

arxiv 2607.29286 v1 pith:AZZGUXZK submitted 2026-07-31 physics.class-ph cond-mat.soft

Revisiting Stress Analysis in a Three-Dimensional Elastic Hollow Sphere under Uniaxial Compression via the Inverse Laplace Transform Expressions within an Elastodynamic Framework

Satoshi Takada , Shintaro Hokada This is my paper

classification physics.class-ph cond-mat.soft

keywords elastic hollow sphereuniaxial compressionelastodynamicsLaplace transformLegendre seriesstress concentrationthin-shell limitclosed-form solution

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.

This paper derives explicit closed-form expressions for the static displacement and stress fields in a three-dimensional elastic hollow sphere squeezed by diametrically opposite point loads. The authors obtain these expressions not by solving the static equations directly but by taking the long-time limit of an elastodynamic solution, using Laplace transforms and expansions in scalar and vector potentials. The payoff is a full, explicit Legendre-series representation of the six displacement and stress components, with coefficients spelled out in the paper. Two concrete results stand out: on the inner surface the radial displacement and stresses scale as 1/(1−ρi) in the thin-shell limit, and the circumferential stress σθθ at the inner surface develops a peak at an intermediate polar angle once the inner radius is large. If correct, these formulas give engineers and materials scientists a directly usable description of stress concentration in porous or hollow spherical components under uniaxial compression.

Core claim

The central claim is that the static response of an elastic hollow sphere under uniaxial compression can be written as explicit infinite series over even Legendre modes, with all coefficients determined by the closed-form rational expressions in Table 2. These are derived by applying the final value theorem to the Laplace-domain elastodynamic solution, so the static formulas inherit a derivation from the full time-dependent equations rather than being fitted or numerically generated. The paper further shows that in the thin-shell limit the inner-surface quantities diverge as (1−ρi)^−1, and that the inner-surface hoop stress σ̃θθ is not monotonic: it develops a peak in 0<θ<π/2 when the inner

What carries the argument

The machinery is a Laplace-domain elastodynamic solution in spherical coordinates. The displacement is written through Helmholtz decomposition into scalar and vector potentials, each expanded as a Legendre series with coefficients built from modified spherical Bessel functions. Enforcing boundary conditions produces a 4×4 linear system whose determinant D_n(s) and numerators N_{n,i}(s) are listed in Table 1. The static limit is then taken by applying the final value theorem term-by-term, converting the Laplace forms into the explicit Legendre series (13a)–(13f) with rational coefficients in the inner radius ρi; the forward/backward (ρi/ρ)^{2n±1} factors encode the influence of the inner free

Load-bearing premise

The paper assumes the final value theorem can be applied term by term to an infinite Legendre series, even though the applied load is a singular point-force distribution and the series is not uniformly convergent near the surfaces.

What would settle it

A direct static numerical simulation of the same boundary-value problem (e.g., finite-element analysis with the same geometry, ν=0.3 and ρi=0.5) should reproduce the inner-surface hoop-stress peak at the same polar angle and magnitude to within the truncation error; if the peak location or the (1−ρi)^−1 scaling disagrees, the termwise final-value interchange is suspect. Additionally, computing the σθθ series at the inner surface with Nmax larger than 1000 would test the stated convergence.

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

If this is right

A direct analytical route from elastodynamics to statics: the same inverse-Laplace machinery can be rerun at finite time to obtain transient wave responses, since the s-domain coefficients D_n(s) and N_{n,i}(s) are already available.
The thin-shell divergence (1−ρi)^−1 quantifies how quickly a hollow sphere's inner-surface displacement and stress grow as the wall gets thinner—useful for failure prediction in thin-walled shells.
The peak of σθθ at an intermediate polar angle identifies where a crack or yield onset would likely initiate on the inner surface for larger inner radii, a qualitative guide for experiments.
Reduction to the solid-sphere limit (ρi→0) recovers known results, providing a consistency check and a single unified series for a family of geometries from solid to thin shell.

Where Pith is reading between the lines

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

Because the tabulated s-domain coefficients are valid for all s, the static series are only one branch of a larger dynamic solution tree; evaluating residues of D_n(s)=0 would yield the natural frequencies and mode shapes of the hollow sphere, which the paper does not compute.
The Legendre basis is fixed to axisymmetric even modes; a natural testable extension is to asymmetric loading (superposed odd-mode pairs) or to a compressed hollow sphere with an eccentric cavity, where the peak location should shift and possibly split.
The claim that σθθ's peak comes from the second term in (13e) suggests a purely geometric criterion: the term decays in θ and grows with ρi, so one can predict the peak angle as the balance point of those two trends without re-summing the full series.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A useful set of explicit static coefficients for the hollow sphere, but the claimed derivation via the final value theorem in an undamped elastodynamic problem is not valid as stated.

read the letter

The paper gives explicit closed-form series for the static displacement and stress fields in an elastic hollow sphere under uniaxial compression. That is genuinely new: Kasano et al. left the coefficients in incremental form, and the solid-sphere limit checks out when ρi→0. The thin-shell scaling and the peak in σθθ at the inner surface are concrete, testable claims. The derivation is standard and the algebra looks plausible, though it is not machine-checked.

Now the soft spots, in proportion.

The biggest one is the derivation route. The paper says the static solutions are obtained in the long-time limit via the final value theorem. But the system is undamped linear elasticity; after a step load, each mode oscillates forever, so the time limit τ→∞ does not exist. The final value theorem requires that all poles of sF(s) lie in the open left half-plane (or at most one at the origin). Here there are imaginary-axis poles, so the theorem does not apply. What the operation lim s→0 sF(s) actually extracts is the residue at s=0, i.e., the time-independent equilibrium component. That can be a valid way to pick out the static solution, but the paper's characterization of it as a long-time limit is wrong as stated. This is not a fatal blow to the formulas themselves—they may well be the correct static equilibrium fields—but the derivation needs to be reframed or justified, perhaps by solving the static problem directly.

Second, the termwise interchange of the limit s→0 with the infinite sum over n is not proved. The boundary condition is a singular point-load expansion, so uniform convergence is not automatic. This is a real gap, but again likely fixable with a careful argument.

Third, the truncation at Nmax=1000 is justified only by a convergence plot for uρ along θ=0. Stresses and points near the surfaces can converge more slowly, and the stress concentration at the inner surface is where the paper makes its main claims. A convergence check for σθθ(ρi,θ) would be reassuring.

Minor: the abstract says the peak occurs where the angle becomes perpendicular (θ=π/2), but the body says the peak sits in the middle of 0<θ<π/2 when ρi is not small. That is a mismatch in wording, not a deep flaw.

Overall: the core result is probably right, and the explicit coefficients are a useful contribution to a classical problem. The derivation needs repair, and the convergence checks need broadening. This deserves a serious referee, not a desk reject. If I were the editor, I would send it out and ask for a revised version that fixes the final-value-theorem issue and tightens the series justification.

Referee Report

3 major / 6 minor

Summary. The paper revisits the static stress analysis of an elastic hollow sphere under uniaxial compression by embedding the problem in an elastodynamic framework. The authors use Helmholtz decomposition and Laplace transforms to derive a Legendre-series solution for the displacement and stress fields, with coefficients expressed through modified spherical Bessel functions. They then apply the final value theorem to obtain static formulas in the long-time limit, giving explicit closed-form coefficients in Table 2. Numerical results show convergence of the radial displacement for N_max=1000, and the paper discusses the inner-surface stress behavior, including a claimed peak in θ_θ at intermediate angles for large inner radii.

Significance. If the derived static expressions are correct, they constitute a useful analytical reference for hollow-sphere stress analysis, complementing classical solid-sphere solutions. The paper's strengths include fully explicit formulas, no fitted parameters, and an internal consistency check by reducing to the solid-sphere limit as ρ_i→0. However, the derivation has a load-bearing gap: the final value theorem is applied to an undamped elastodynamic system with purely imaginary poles, so the claimed long-time limit is not justified. The convergence of the truncated series is also only demonstrated for u_ρ, not for the stress components. These issues are significant but appear fixable within the manuscript's scope.

major comments (3)

[Section 3, Eqs. (12)-(13)] The final value theorem is misapplied. The system is undamped linear elasticity; the step-loaded response contains oscillatory terms with imaginary-axis poles, so lim_{τ→∞} of the solution does not exist termwise. The final value theorem requires all poles of sF(s) to lie in the open left half-plane, except possibly one at the origin. The operation lim_{s→0} sF(s) can only extract the residue at s=0, i.e., the time-independent equilibrium component, not the long-time limit. Thus the derivation of Eqs. (13) as 'long-time limits' is unsupported. Please revise Section 3: either derive the static solutions directly from the elastostatic equations, or explicitly state that you are taking the s=0 residue to obtain the static equilibrium component (and justify the termwise limit).
[Section 3, Fig. 2 and Section 4] The truncation N_max=1000 is justified only by the convergence of ũ_ρ^{(st)}(ρ,0) shown in Fig. 2. The stress components, particularly near the surfaces ρ=1 and ρ=ρ_i, have series with ρ^{n-2} factors and the singular point-load boundary condition (1a) makes nonuniform convergence near the loading points very likely. The claim that N_max=10^3 is sufficient for all displayed quantities is not supported. Provide convergence tests for the stress components at representative points (e.g., inner surface at θ=0 and θ=π/2, outer surface at θ=0) as a function of N_max, or give an a priori estimate of the rate of decay of the terms.
[Section 3, Table 2] The jump from the Laplace-domain coefficients in Table 1 to the static coefficients in Table 2 is stated only as 'After some calculations'. Since the explicit closed-form expressions are the paper's central contribution, the derivation should be verifiable. At minimum, include an outline of the limit s→0 and the algebraic manipulation of D_n(s) and N_{n,i}(s), or provide a supplementary file with the full derivation. This is especially important given the questionable applicability of the final value theorem.

minor comments (6)

[Title] 'REVISTING' should be 'REVISITING'.
[Section 2, Eq. (12)] The inverse Laplace transform formula is typeset as '1/(2π/i)' and the integration limits are given in reverse order; it should read 1/(2π i) ∫_{γ-i∞}^{γ+i∞}.
[Section 1, Eq. (1a)] The sum over even n is a representation of two diametrically opposed point loads at θ=0 and θ=π. A brief explanation would improve readability.
[Abstract] The abstract says quantities peak 'at the point where the angle ... becomes perpendicular' (θ=π/2), but the paper's own result (Fig. 5) shows the σθθ peak at an intermediate angle, not exactly θ=π/2. Please align the abstract with the actual results.
[References] Reference [12] is a Japanese-language paper; consider citing an accessible English version or providing a translation note.
[Fig. 2] The label 'converge' should be 'convergence'.

Circularity Check

0 steps flagged

No significant circularity: the static closed-form solution is obtained from the stated boundary-value problem by Laplace transform and the s→0 limit, with no fitted parameters or load-bearing self-citations.

full rationale

The paper's central derivation is self-contained. The governing equation (3), Helmholtz decomposition (5), Laplace-domain potentials (7), and boundary conditions (1) are the inputs; the coefficients (11) and Tables 1–2 are obtained by imposing those boundary conditions, not by fitting to any output quantity. The static formulas (13a)–(13f) are then obtained as the s→0 limit via the final value theorem. There is no step in which the derived quantity is defined in terms of the predicted quantity, no fitted parameter is renamed as a prediction, and no uniqueness claim is imported from the authors' prior work. The self-citations to [11,12] and [5,12] are used only as consistency checks ('the expressions reduce to those for solid spheres [5, 12] when we consider ρi→0') and as references for residue-evaluation techniques; they do not carry the derivation. The correctness concern that the final value theorem may not be termwise valid for the undamped infinite Legendre series is a validity/mathematical-rigor issue, not a circularity issue: even if that step were unsupported, the static fields do not reduce by construction to the dynamic input. Overall, the paper shows no significant circularity; the score reflects only the presence of self-citations that are not load-bearing.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities and no fitted free parameters. Its load-bearing assumptions are the standard linear-elastic model, the point-load idealization of the boundary condition, and two analytical short-cuts (termwise final-value theorem and truncation at Nmax=1000) that are asserted without rigorous justification in the manuscript.

axioms (4)

domain assumption The material is homogeneous, isotropic, and linear elastic, governed by the Navier-Cauchy equation and Hooke's law.
Eqs. (3) and (10) are adopted without discussion of their range of validity; this is a standard modeling assumption for small deformations.
domain assumption The uniaxial compression is represented by the singular Legendre series in Eq. (1a), corresponding to point loads at the poles θ=0 and θ=π.
This is a standard idealization for diametral compression tests, but the singular nature of the loading affects the convergence of the Legendre series.
ad hoc to paper The final value theorem can be applied termwise to the infinite Legendre series of Laplace-transformed coefficients to obtain the static limit.
The interchange of lim_{s→0} and Σ_n is not proved; the series is not uniformly convergent near ρ=1 because of the singular boundary data.
ad hoc to paper Truncating the series at Nmax=1000 gives sufficiently accurate results for all displayed quantities, including stresses near the surfaces.
Convergence is only demonstrated for uρ in Fig. 2; no error bounds are given for stress components near the inner or outer surface.

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read the original abstract

The stress analysis of a three-dimensional elastic hollow sphere subjected to uniaxial compression is revisited, employing an elastodynamic framework. Through the application of the Laplace transform, the scalar and vector potentials of displacement are expanded, facilitating a detailed exploration of the system's mechanical behavior. The static solutions for displacement and stress distributions are derived in the long-time limit, which reveal key insights into the response of the elastic hollow sphere. Notably, on the inner surface, certain quantities exhibit a peak at the point where the angle between the compressive force and the point on the surface becomes perpendicular, indicating localized stress concentration. These findings provide a robust analytical approach for understanding and predicting the behavior of elastic hollow spheres under uniaxial loading, with implications for material science and structural engineering.

Figures

Figures reproduced from arXiv: 2607.29286 by Satoshi Takada, Shintaro Hokada.

**Figure 1.** Figure 1: A schematic of the system. An elastic hollow sphere whose outer and inner radii are 𝑅o and 𝑅i , respectively, is compressed uniaxially (𝜃 = 0 and 𝜋) with the pressure 𝜎0. The boundary conditions are given by 𝜎𝑟𝑟|𝑟=𝑅o = −𝜎0Θ(𝑡) ∑ (2𝑛 + 1)𝑃𝑛 (cos 𝜃) ∞ 𝑛=0,2,4,⋯ , (1a) 𝜎𝑟𝑟|𝑟=𝑅i = 𝜎𝑟𝜃|𝑟=𝑅o = 𝜎𝑟𝜃|𝑟=𝑅i = 0, (1b) where 𝑃𝑛 (𝑥) stands for the Legendre polynomial [15]. It is convenient to introduce the following sca… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages

[1]

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

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Y. C. Fung and P. Tong, Classical and Computational Solid Mechanics, World Scientific (2001)

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L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity, Elsevier (1986)

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A. C. Eringen and E. S. Şuhubi, Elastodynamics, Vol. II Linear Theory, Academic Press (1975)

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Hiramatsu and Y

Y. Hiramatsu and Y. Oka, Determination of the tensile strength of rock by a compression test of an irregular test piece, Int J Rock Mech Min Sci Abst, 3, 89–90 (1966)

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Ramesh and K

K. Ramesh and K. Shins, Stress field equations for a disk subjected to self -equilibrated arbitrary loads: revisited, Granul Matter, 24, 49 (2022)

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[7]

Shins and K

K. Shins and K. Ramesh, Closed-form displacement field equations for a disc subjected to self -equilibrated arbitrary loads, Granul Matter, 25, 38 (2023)

2023
[8]

Jingu, K

T. Jingu, K. Hisada, I. Nakahara, and S. Machida, Transient stress in a circular disk under diametrical impact loads, Bulletin JSME, 28, 13–23 (1985)

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[9]

Jingu and K

T. Jingu and K. Nezu, Transient Stress in an Elastic Sphere under Diametrical Concentrated Impact Loads , Bulletin JSME, 28, 2553–2561 (1985)

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[10]

S. Z. Wu and K. T. Chau, Dynamic response of an elastic sphere under diametral impacts , Mech Mater, 38, 1039–1060 (2006)

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[11]

Y. Sato, H. Ishikawa, and S. Takada, Revisiting stress propagation in a two-dimensional elastic circular disk under diametric loading, J Elast, 156, 193–213 (2024)

2024
[12]

Sato and S

Y. Sato and S. Takada, Revisiting stress propagation in a three -dimensional elastic sphere under diametric loading, Trans JSME (in Japanese), 90, 23-00262 (2024)

2024
[13]

Kasano, I

H. Kasano, I. Nakahara, H. Matsumoto, and E. Tsuchida, Trans JSME (in Japanese), 41, 2847–2852 (1975). (https://doi.org/10.1299/kikai1938.41.2847)

work page doi:10.1299/kikai1938.41.2847 1975
[14]

Okamura, Y

K. Okamura, Y. Sato, and S. Takada, Displacement and Stress Analysis of an Elastic Hollow Disk: Comparison with Strength of Materials’ Prediction, Strength Mater, 57, 102–114 (2025)

2025
[15]

Abramowitz and I

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover (1964)

1964
[16]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge Univ. Press (2007)

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[17]

Yokoyama, B

Y. Yokoyama, B. R. Mitchell, A. Nassiri, B. L. Kinsey, Y. P. Korkolis, and Y. Tagawa, Integrated photoelasticity in a soft material: phase retardation, azimuthal angle, and stress -optic coefficient, Opt Lasers Eng, 161, 107335 (2023)

2023

[1] [1]

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

1970

[2] [2]

Y. C. Fung and P. Tong, Classical and Computational Solid Mechanics, World Scientific (2001)

2001

[3] [3]

L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity, Elsevier (1986)

1986

[4] [4]

A. C. Eringen and E. S. Şuhubi, Elastodynamics, Vol. II Linear Theory, Academic Press (1975)

1975

[5] [5]

Hiramatsu and Y

Y. Hiramatsu and Y. Oka, Determination of the tensile strength of rock by a compression test of an irregular test piece, Int J Rock Mech Min Sci Abst, 3, 89–90 (1966)

1966

[6] [6]

Ramesh and K

K. Ramesh and K. Shins, Stress field equations for a disk subjected to self -equilibrated arbitrary loads: revisited, Granul Matter, 24, 49 (2022)

2022

[7] [7]

Shins and K

K. Shins and K. Ramesh, Closed-form displacement field equations for a disc subjected to self -equilibrated arbitrary loads, Granul Matter, 25, 38 (2023)

2023

[8] [8]

Jingu, K

T. Jingu, K. Hisada, I. Nakahara, and S. Machida, Transient stress in a circular disk under diametrical impact loads, Bulletin JSME, 28, 13–23 (1985)

1985

[9] [9]

Jingu and K

T. Jingu and K. Nezu, Transient Stress in an Elastic Sphere under Diametrical Concentrated Impact Loads , Bulletin JSME, 28, 2553–2561 (1985)

1985

[10] [10]

S. Z. Wu and K. T. Chau, Dynamic response of an elastic sphere under diametral impacts , Mech Mater, 38, 1039–1060 (2006)

2006

[11] [11]

Y. Sato, H. Ishikawa, and S. Takada, Revisiting stress propagation in a two-dimensional elastic circular disk under diametric loading, J Elast, 156, 193–213 (2024)

2024

[12] [12]

Sato and S

Y. Sato and S. Takada, Revisiting stress propagation in a three -dimensional elastic sphere under diametric loading, Trans JSME (in Japanese), 90, 23-00262 (2024)

2024

[13] [13]

Kasano, I

H. Kasano, I. Nakahara, H. Matsumoto, and E. Tsuchida, Trans JSME (in Japanese), 41, 2847–2852 (1975). (https://doi.org/10.1299/kikai1938.41.2847)

work page doi:10.1299/kikai1938.41.2847 1975

[14] [14]

Okamura, Y

K. Okamura, Y. Sato, and S. Takada, Displacement and Stress Analysis of an Elastic Hollow Disk: Comparison with Strength of Materials’ Prediction, Strength Mater, 57, 102–114 (2025)

2025

[15] [15]

Abramowitz and I

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover (1964)

1964

[16] [16]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge Univ. Press (2007)

2007

[17] [17]

Yokoyama, B

Y. Yokoyama, B. R. Mitchell, A. Nassiri, B. L. Kinsey, Y. P. Korkolis, and Y. Tagawa, Integrated photoelasticity in a soft material: phase retardation, azimuthal angle, and stress -optic coefficient, Opt Lasers Eng, 161, 107335 (2023)

2023