🔬 physics

Analysis of Stresses and Displacements in Two-Layer Inhomogeneous Hollow Cylinders Using the Complex Potential Method

This study presents an analytical solution for the thermoelastic state of a long, two-layer hollow cylinder with inhomogeneous materials and internal heat generation under mixed boundary conditions by utilizing the Kolosov–Muskhelishvili complex potential method and power series expansions to derive precise formulas for stress and displacement distributions.

Original authors: Youssri Hassan Youssri, Zafar Duman Abbasov, Lamine Bouzettouta, Amal Alhujaylan, Khaled Zennir

Published 2026-09-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Youssri Hassan Youssri, Zafar Duman Abbasov, Lamine Bouzettouta, Amal Alhujaylan, Khaled Zennir

Original paper licensed under CC BY 4.0 (https://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 a world where the materials holding up our bridges, the pipes carrying oil through the ocean, and the engines powering spacecraft are subjected to extreme heat and pressure. In these high-stakes environments, engineers often rely on structures made of multiple layers, like a hollow cylinder composed of two different materials joined together. The challenge arises when these layers are not uniform; their properties change from the inside out, and they are subjected to internal heat sources that vary in intensity. When heat is applied, materials expand. If the layers expand at different rates or if the heat is uneven, immense stress builds up at the junction where the materials meet. If this stress is not understood and managed, the structure can crack, warp, or fail catastrophically. For decades, predicting exactly how these complex, layered cylinders behave under such thermal loads has been a difficult puzzle, often requiring rough approximations or expensive computer simulations that struggle to capture the precise details at the interface between layers.

A team of researchers has now solved this puzzle with a new, exact mathematical approach. They focused on a long, hollow cylinder made of two distinct layers of material that are not uniform throughout. Unlike previous studies that might have treated the materials as simple or uniform, this work accounts for the fact that the material properties change and that heat is being generated inside the structure itself. The researchers developed a precise analytical solution, meaning they derived a set of exact formulas that describe exactly how the cylinder will stretch, compress, and shift when heated. They did this by translating the physical problem of heat and stress into the language of complex numbers, a powerful mathematical tool that simplifies the geometry of circles and rings. By expressing the temperature and the resulting movement of the material as infinite series of terms, they were able to calculate the exact stress and displacement at every point within the cylinder, from the inner surface to the outer edge.

The study reveals that the intensity of the internal heat source plays a massive role in determining the stress field inside the cylinder. The researchers found that even small changes in how heat is generated or distributed can significantly alter the forces acting on the material. Their method allows them to see exactly where the stress concentrates, particularly at the contact surface where the two different layers meet. They demonstrated that their formulas work whether the surfaces are insulated to keep heat in or kept at a specific temperature to keep heat out. This level of precision is crucial because it provides a reliable benchmark. Engineers can now use these exact formulas to check the accuracy of their computer simulations, ensuring that the designs for aerospace vehicles, nuclear reactors, and deep-sea pipelines are safe and durable.

To prove their new method works, the team also built a robust numerical framework, a type of high-precision calculation method, to test their formulas against known scenarios. They simulated a cylinder with an inner radius of 1 meter, an interface at 2 meters, and an outer radius of 3 meters. In their tests, they used materials with specific properties, such as a thermal conductivity that varied from 1 plus 0.5 times the radius in the first layer to 2 minus 0.3 times the radius in the second. They applied a heat source that varied smoothly across the layers. The results were striking: their method achieved an accuracy so high that the error was reduced to nearly zero, reaching a level of precision that standard computer methods struggle to match without using vastly more computing power. For instance, to reach a certain level of accuracy, their method required far fewer calculation points than traditional approaches, making it much faster and more efficient.

The researchers also tested how their method held up when the materials changed abruptly, such as when the thermal conductivity jumped by a factor of ten at the interface. Even in these difficult cases, where the material properties are discontinuous, their approach maintained high accuracy. This is a significant finding because real-world engineering often involves sharp transitions between different materials. The study confirms that their analytical solution is stable and reliable, meaning that small changes in the heat input or material properties lead to predictable, small changes in the stress results. This reliability is essential for designing structures that must withstand the unpredictable thermal environments of space or deep-sea operations.

Ultimately, this work provides a sophisticated tool for modern engineering. By offering a clear, exact picture of how heat and stress interact in complex, layered cylinders, the researchers have filled a gap in the scientific literature. Their findings are not just theoretical; they offer a practical way to optimize the design of multi-layered structures, ensuring they can resist the thermal loads they will face in the real world. The ability to predict these stresses with such precision means that engineers can design safer, more efficient, and more durable components for the high-technology fields that drive our modern society.

Technical Summary: Analysis of Stresses and Displacements in Two-Layer Inhomogeneous Hollow Cylinders Using the Complex Potential Method

Problem Statement
This study addresses the thermoelastic state of a long, two-layer hollow cylinder composed of inhomogeneous materials. The cylinder is subjected to mixed boundary conditions and internal heat generation. Specifically, the research focuses on a composite ring where distinct thermoelastic properties exist within each layer, and a heat source of intensity $W(\theta)$ acts along the contact circumference ( $r=r_1$ ). The boundary conditions involve surfaces that are either thermally insulated or maintained at zero temperature, while the interface requires continuity of temperature and a specific jump in heat flux corresponding to the internal source. The mathematical challenge lies in solving the coupled system of differential equations governing the thermal and elastic fields in an inhomogeneous medium with mixed boundary constraints, a scenario where traditional numerical methods often struggle with accuracy near material interfaces.

Methodology
The paper employs a rigorous analytical approach based on the classical Kolosov–Muskhelishvili complex potential method. The solution procedure is structured as follows:

Temperature Field Determination: The temperature distribution $T^{(k)}(r, \theta)$ is derived by solving the stationary heat conduction equation. The solution is expressed as a power series expansion, where the temperature function is represented as a sum of harmonic terms involving radial functions. This allows for the handling of material inhomogeneities and the specific heat source distribution.
Complex Potential Formulation: The thermoelastic problem is transformed into the complex domain using Kolosov–Muskhelishvili potentials, $\phi_k(z)$ and $\psi_k(z)$ . These potentials are expressed as Laurent series expansions ( $\sum A_n^{(k)} z^n$ and $\sum B_n^{(k)} z^n$ ).
Stress and Displacement Calculation: Using the established potentials and the derived temperature field, the radial and tangential stresses ( $\sigma_r, \sigma_\theta$ ) and displacements ( $u_r, u_\theta$ ) are calculated via standard complex variable formulas. The particular solution for the thermal displacements is sought in the form of a Laurent series to ensure analytical convergence across the annular domains.
Coefficient Determination: The unknown coefficients in the series expansions are determined by enforcing boundary conditions (zero stress on outer/inner surfaces) and contact conditions (continuity of displacement and stress jumps at the interface). This results in an infinite system of linear algebraic equations, which is solved to determine the unique coefficients for the potentials.
Mathematical Rigor: The paper establishes the existence, uniqueness, and stability of the solution through a series of lemmas and theorems. It proves that the solution is well-posed in the sense of Hadamard and demonstrates that the stress and displacement fields are continuous within layers and stable with respect to perturbations in the heat source.
Numerical Validation: To validate the analytical results, the authors developed a numerical framework using the Shifted Jacobi Collocation Method (SJCM). This spectral method was applied to the axisymmetric case to verify the convergence and accuracy of the analytical solution.

Key Contributions and Results

Analytical Solution: The primary contribution is the derivation of an exact analytical solution for the thermoelastic state of a two-layer inhomogeneous hollow cylinder under mixed boundary conditions. This fills a gap in the literature regarding the application of complex potential methods to such specific composite geometries.
Closed-Form Expressions: The study provides closed-form expressions for stress and displacement components, explicitly accounting for the intensity of the internal heat source and the inhomogeneity of the materials.
Numerical Verification: The SJCM validation demonstrates spectral accuracy. For smooth solutions, the method achieves machine precision (errors $\approx 10^{-15}$ ) with a relatively low polynomial degree ( $N \approx 64$ ). The numerical results confirm that the analytical solution is stable and that the method handles material discontinuities effectively when using domain decomposition.
Parameter Sensitivity: The numerical experiments identify $(\alpha, \beta) = (0, 1)$ as the optimal Jacobi parameters for radial problems involving singular coefficients (like $1/r$ ), yielding the smallest errors and best conditioning.

Significance and Claims
The paper claims significant theoretical and practical value for the following reasons:

Theoretical Novelty: It represents the first implementation of the Kolosov–Muskhelishvili complex potential method specifically for determining thermoelastic stresses and displacements in two-layer inhomogeneous hollow cylinders. Unlike simplified homogeneous models, this approach rigorously accounts for material inhomogeneity and non-uniform heat sources at the contact interface.
Engineering Application: The derived formulas provide a robust mathematical framework for high-precision engineering applications. The results are particularly significant for the design and durability assessment of multilayered structural elements in aerospace, high-precision liners, satellite systems, and shipbuilding, where resistance to external thermal loads is critical.
Benchmarking: The exact analytical solution serves as a rigorous benchmark for validating numerical simulations, offering deeper physical insight into the coupling between thermal and elastic fields in complex layered structures.
Stability: The study demonstrates that the internal stress field is significantly influenced by the intensity of the internal heat source, and the proposed analytical model is stable against small perturbations in thermal loading, ensuring reliability for practical engineering calculations.

The authors conclude that the proposed methodology successfully bridges the gap between thermal potential fields and mechanical responses, offering a sophisticated tool for modern engineering analysis of layered cylinders under diverse thermal environments.

More like this