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

This paper revisits the stress analysis of a three-dimensional elastic hollow sphere under uniaxial compression using an elastodynamic framework and Laplace transform techniques, deriving static solutions that reveal localized stress concentrations on the inner surface perpendicular to the applied load.

Original authors: Satoshi Takada, Shintaro Hokada

Published 2026-08-03

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CC BY 4.0

Original authors: Satoshi Takada, Shintaro Hokada

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of materials science as a giant, invisible playground where everything from the bridge you drive over to the bone in your knee is made of tiny, stretchy springs. When you push or pull on these structures, they wiggle, squash, and stretch in ways that follow strict rules called "elasticity." Think of it like a trampoline: if you jump in the middle, the fabric dips down, but if you push on the edge, the whole thing ripples in a specific pattern. Scientists have spent a century figuring out exactly how solid balls of material react when you squeeze them, which helps engineers build safer buildings and better tools. But here's the twist: real life isn't perfect. Most materials aren't solid blocks; they are full of tiny holes, bubbles, or empty spaces, kind of like a sponge or a hollowed-out orange. These empty spots change how the material handles stress, often creating weak spots where things might crack. Understanding how these "hollow" objects behave is like trying to predict how a hollow rubber ball will squish when you squeeze it, which is much trickier than figuring out a solid one.

This paper takes a fresh look at a classic problem: what happens when you squeeze a hollow, elastic sphere (like a perfect, hollow rubber ball) from the top and bottom? The authors, S. Takada and S. Hokada, decided to revisit this using a powerful mathematical toolkit called "elastodynamics," which is basically the study of how things move and shake when forces are applied. Instead of just guessing or using messy, step-by-step computer approximations, they used a clever math trick called the "Laplace transform" to break the problem down into simpler pieces. They treated the hollow sphere like a musical instrument, where the vibrations (or in this case, the stress) can be described as a mix of different notes (mathematical waves). By solving the equations for these waves, they managed to write down exact, clear formulas for how the sphere stretches and where the stress builds up.

The big discovery here is that the hollow sphere behaves in some surprising ways compared to a solid one. When the researchers looked at the inside surface of the hollow ball, they found that the stress doesn't just spread out evenly. Instead, it creates a "hotspot" or a peak of pressure at the very spot on the inside wall that is perpendicular to the squeezing force. Imagine squeezing a hollow ball between your hands; the inside wall right in the middle of the sides (not where your hands are touching) suddenly feels a sharp spike in pressure. The paper shows that as the hole in the middle gets bigger (making the shell thinner), this pressure spike gets even more intense, growing roughly in proportion to how thin the shell is. They also found that while the pressure pushing inward drops to zero near the inner hole, the pressure pulling outward (tension) actually gets stronger as you get closer to the inner edge.

The authors are very confident in these results because they derived them using strict mathematical proofs, not just computer simulations. They showed that their formulas work perfectly by checking how they behave when the hole is tiny (turning it back into a solid sphere) and when the hole is huge (making it a thin shell). They even proved that their math converges, meaning if you add up enough of the "notes" in their formula, the answer settles down to a precise, stable number. While they didn't test this on a physical rubber ball in a lab, their mathematical framework is so robust that it can be used to predict exactly how any hollow sphere will react, whether it's a tiny ceramic bead or a large structural component. This work gives engineers a new, precise map for understanding where hollow materials might fail, helping them design safer structures that can handle the squeeze without cracking.

Technical Summary: Revisiting Stress Analysis in a Three-Dimensional Elastic Hollow Sphere Under Uniaxial Compression

Problem Statement
This study addresses the stress analysis of a three-dimensional elastic hollow sphere subjected to uniaxial compression. While the behavior of solid spheres under similar loading is well-documented in classical elasticity theory, and dynamic analyses exist for cylinders and spheres, there remains a gap in the literature regarding comprehensive, explicit closed-form solutions for hollow spheres. Previous work, such as that by Kasano et al., addressed hollow spheres but determined displacement and stress coefficients in an incremental form, lacking the clarity and utility of explicit expressions. This paper aims to fill that gap by deriving explicit analytical expressions for the displacement and stress coefficients, thereby enhancing the theoretical understanding of stress behavior in porous or hollow spherical structures.

Methodology
The authors employ an elastodynamic framework to model the system. The problem is defined by an elastic hollow sphere with outer radius $R_o$ and inner radius $R_i$ , characterized by mass density $\varrho_0$ , shear modulus $G$ , and Poisson's ratio $\nu$ . The sphere is subjected to uniaxial compression defined by a pressure $\sigma_0$ acting diametrically.

The solution process involves the following steps:

Governing Equations: The dimensionless Navier-Cauchy equation is solved using the Helmholtz theorem, which decomposes the displacement vector into scalar ( $\varphi$ ) and vector ( $\chi$ ) potentials.
Laplace Transform: The potentials are transformed into the Laplace domain. The solutions are expanded using modified spherical Bessel functions of the first and second kinds ( $i_n$ and $k_n$ ) and Legendre polynomials ( $P_n$ ).
Boundary Conditions: The coefficients of the expansion are determined by satisfying the boundary conditions at the inner and outer surfaces. This yields a system of simultaneous equations, the solution of which provides explicit expressions for the coefficients $a_n, b_n, c_n,$ and $d_n$ in terms of determinants involving the Bessel functions and material parameters.
Static Limit: To obtain static solutions, the authors apply the final value theorem to the inverse Laplace transform expressions. This allows the derivation of explicit infinite series for the static displacement and stress components as the time parameter approaches infinity.
Numerical Implementation: The convergence of the infinite series is analyzed, establishing that a truncation limit of $N_{max} = 1000$ is sufficient for accurate computation, even near the inner surface.

Key Results
The study derives explicit series expressions for the dimensionless radial and tangential displacements ( $\tilde{u}_\rho, \tilde{u}_\theta$ ) and stress components ( $\tilde{\sigma}_{\rho\rho}, \tilde{\sigma}_{\rho\theta}, \tilde{\sigma}_{\theta\theta}, \tilde{\sigma}_{\phi\phi}$ ). Key findings include:

Stress Concentration: The magnitude of deformation and the second principal stress difference are enhanced in the vicinity of the loading points (poles).
Comparison with Solid Spheres: On the loading axis ( $\theta=0$ ), the compression stress in a hollow sphere behaves similarly to a solid sphere in the bulk region but decreases sharply as the inner surface is approached. Conversely, the extension stress is lower than that of a solid sphere in the intermediate region but increases significantly near the inner diameter.
Inner Surface Behavior: At the inner surface ( $\rho = \rho_i$ ), the circumferential stress $\tilde{\sigma}_{\theta\theta}$ exhibits a distinct peak at an angle $\theta$ between 0 and $\pi/2$ (specifically where the angle between the compressive force and the surface point becomes perpendicular). This peak becomes more pronounced as the inner radius increases.
Thin Shell Limit: In the limit where the shell width is small ( $\delta \ll 1$ ), the radial displacement and stress components scale approximately as $(1 - \rho_i)^{-1}$ , consistent with thin-shell theory expectations.

Significance and Claims
The authors claim that this work provides a robust analytical approach for understanding and predicting the mechanical behavior of elastic hollow spheres under uniaxial loading. By moving from incremental to explicit closed-form expressions, the study offers practical tools for:

Simplifying the computational modeling of porous structures.
Aiding in the design of materials with tailored mechanical properties.
Improving the accuracy of failure predictions in engineering applications involving hollow or porous spherical geometries.

The paper concludes that these findings contribute to a deeper understanding of stress distribution in complex internal geometries. The authors note that while the current focus is on static solutions, the derived inverse Laplace transform expressions can be extended to dynamic problems to analyze wave propagation phenomena (longitudinal, transverse, and Rayleigh waves), though such dynamic analysis would require complex residue evaluation and case categorization based on time and location.

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