🔬 physics

Hyperelastic characterization of a cylinder under tension -- torsion

This study presents a theoretical framework based on the Pucci–Saccomandi model to analyze the nonlinear mechanical response of an incompressible hyperelastic cylinder under combined tension and torsion, revealing pronounced strain-stiffening, strong axial-twist coupling, and the potential for Poynting effect reversal depending on material parameters.

Original authors: MADAHAN BIEN--AIME LIMAN KAOYE, Blaise BALE B., GAMBO BETCHEWE

Published 2026-08-03

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: MADAHAN BIEN--AIME LIMAN KAOYE, Blaise BALE B., GAMBO BETCHEWE

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 materials don't just stretch like silly putty or snap like dry twigs, but dance to a complex, invisible rhythm. This is the realm of hyperelasticity, the study of materials like rubber that can undergo massive, squishy deformations and bounce right back. To understand how these materials behave, scientists use "constitutive models," which are essentially mathematical recipes. These recipes predict how a material will react when you pull it, twist it, or squeeze it. One of the most fascinating quirks in this world is the Poynting effect. Picture twisting a rubber band: instead of just spinning, the band actually tries to get longer or shorter along its length, as if it's fighting the twist. This happens because the material's internal structure gets tangled and stressed in a way that pushes it to change shape in a direction you didn't expect. Engineers care deeply about this because rubber isn't just for erasers; it's in car tires, airplane parts, and medical devices. If you don't understand how rubber twists and stretches at the same time, your designs might fail when the real world gets messy.

This paper dives deep into that exact messiness. The authors, a team of researchers from universities in Cameroon, decided to play with a mathematical "rubber cylinder" to see how it behaves when you pull it and twist it simultaneously. They didn't just guess; they used a super-smart computer search tool called a genetic algorithm. Think of this algorithm like a digital evolution lab: it creates thousands of random "recipes" for rubber, tests them against real-world data from a famous experiment by Rivlin and Saunders, and then breeds the best ones together, keeping the traits that worked and tossing the ones that failed, until it finds the perfect recipe.

The team tested two different mathematical recipes for this rubber. The first was the Pucci–Saccomandi model, which is a sophisticated upgrade of a classic recipe called the Gent model. The second was a newer recipe by Liman et al., which tweaked the math to handle the twisting part differently. When they ran their simulations, both recipes were great at predicting how much torque (the twisting force) was needed to turn the cylinder. They both matched the experimental data almost perfectly, like two different singers hitting the same high note.

However, the real drama happened when they looked at the axial force—the push or pull along the length of the cylinder caused by the twist (the Poynting effect). Here, the two recipes started to disagree. The Pucci–Saccomandi model tended to slightly underestimate how hard the rubber pushed back, especially when the twist was strong. The Liman et al. model, on the other hand, predicted a stronger push that matched the real-world experiments much better. The authors found that the Liman model was more accurate because it handled the interaction between the material's stretching and twisting in a more clever way.

The paper also explored what happens when you stretch the cylinder while twisting it. They discovered that stretching the rubber first makes the Poynting effect much more dramatic. It's like pulling a rubber band tight before twisting it; the band fights back with much more intensity. The authors suggest that the way the material's internal structure resists this combined stress is the key to understanding why the newer model works better. While the study confirms that the Liman model is currently the superior choice for predicting these complex behaviors, the authors note that this is a theoretical and simulation-based victory. They suggest that future work could test these ideas on different shapes, like hollow tubes, or add real-world complications like how rubber gets tired over time. For now, though, they've given engineers a sharper tool to predict how rubber will behave when life gets complicated.

Technical Summary: Hyperelastic Characterization of a Cylinder under Tension–Torsion

Problem Statement
This study addresses the mechanical characterization of incompressible, rubber-like materials subjected to combined tension and torsion loading. While hyperelastic constitutive models are well-established for uniaxial and shear deformations, combined loading conditions remain less explored despite their relevance in engineering applications such as automotive, aerospace, and biomedical systems. Specifically, the paper investigates the nonlinear mechanical response of a cylinder, focusing on the coupling between axial stretch and twist, and the resulting Poynting effect (the emergence of an axial force under pure torsion). The research aims to evaluate the predictive capabilities of specific hyperelastic models in capturing these complex multiaxial interactions.

Methodology
The authors develop a theoretical framework based on finite kinematics for a homogeneous, incompressible, isotropic cylinder.

Kinematics: The deformation is described using principal stretches and the right Cauchy–Green tensor invariants ( $I_1$ and $I_2$ ). The deformation mapping accounts for an axial stretch ratio ( $\lambda$ ) and a twist per unit length ( $\psi$ ).
Constitutive Modeling: Two specific hyperelastic strain-energy functions are analyzed:
1. The Pucci–Saccomandi model, which extends the Gent model by adding a logarithmic term dependent on $I_2$ .
2. The Liman et al. model, which modifies the Gent framework by introducing a square-root dependency on $I_2$ to better predict biaxial responses.
Analytical Derivation: General expressions for the Cauchy stress tensor, axial force ( $N$ ), and torque ( $M$ ) are derived by integrating stress components over the cylinder's cross-section. Analytical approximations for $N$ and $M$ are obtained via Taylor expansion with respect to the twist angle.
Parameter Identification: To validate the models, material parameters ( $\mu$ , $J_m$ , and $K$ ) are identified using a Genetic Algorithm (GA). The optimization minimizes the mean squared error between the models' predictions and the experimental data from Rivlin and Saunders [35], a standard benchmark for rubber torsion.

Key Contributions

Derivation of Resultant Quantities: The paper provides explicit analytical expressions for the axial force and torque under combined tension–torsion, detailing the contributions of the first and second invariants ( $I_1$ and $I_2$ ) and their radial distributions.
Comparative Model Analysis: By applying the same optimization procedure to two distinct formulations, the study isolates the influence of the functional form of the $I_2$ term (logarithmic vs. power-law/square-root) on the material's response.
Poynting Effect Characterization: The work analyzes the magnitude and behavior of the Poynting effect, demonstrating how the choice of constitutive parameters and the functional form of the strain energy function influence the axial force generated during torsion, including the predicted possibility of a reversal of the effect depending on parameter selection.

Results

Torsional Response: Both the Pucci–Saccomandi and Liman et al. models demonstrate excellent agreement with the Rivlin–Saunders experimental data regarding the torsional moment ( $M$ ) across the deformation range, with coefficients of determination ( $R^2$ ) exceeding 0.999.
Axial Force Prediction: Significant differences emerge in the prediction of the normal force ( $N$ ). The Liman et al. model provides a more accurate fit to the experimental data, particularly at higher deformation levels, exhibiting lower Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) compared to the Pucci–Saccomandi model.
Coupling Effects: The results confirm a strong nonlinear coupling between axial stretch and twist. Increasing the axial stretch ratio ( $\lambda$ ) leads to a marked amplification of the axial force, reflecting strain-stiffening behavior.
Parameter Sensitivity: The study reveals that the Pucci–Saccomandi model tends to slightly overestimate the response, while the Liman et al. model remains in closer agreement, suggesting that the square-root dependency on $I_2$ better captures the nonlinear coupling mechanisms governing the normal force.

Significance and Claims
The paper claims to provide a consistent theoretical basis for the experimental identification of constitutive parameters from combined loading tests. It asserts that the proposed framework effectively captures complex nonlinear interactions in elastomeric materials. The primary finding is that while both models are suitable for describing torsional responses, the Liman et al. formulation offers superior accuracy for the axial force component, which is more sensitive to hydrostatic effects and nonlinear deformation mechanisms. The authors conclude that the functional form of the second invariant ( $I_2$ ) is critical for accurately modeling elastomers under finite multiaxial deformations. The study positions itself as a useful benchmark for the comparative assessment of hyperelastic models under coupled deformation modes, without claiming to solve all open problems in the field or proposing new experimental protocols beyond the validation against existing literature.

More like this