Asymptotic analysis of torsional buckling in nearly incompressible hyperelastic cylinders

Torsion of hyperelastic cylinders exhibits the classical Poynting effect and may lead to torsional buckling when the twist is sufficiently large. Since most elastomeric materials are nearly incompressible, the critical buckling torque is often highly sensitive to small deviations from perfect incompressibility. In this work, we investigate the torsional buckling of a solid circular cylinder composed of a compressible Mooney--Rivlin material and develop an asymptotic expansion of the critical twist in the nearly incompressible limit. Starting from the exact finite deformation associated with uniform torsion and fixed end-to-end distance, we derive the pre-stress field and formulate the incremental boundary-value problem in the current configuration. The incremental equations are cast in Stroh form for a seven-dimensional state vector collecting displacements, tractions, and the pressure increment. The resulting radial eigenvalue problem defines a differential operator $\cL(\kappa,\varepsilon)$, where $\kappa$ is a non-dimensional twist parameter and $\varepsilon$ is a small parameter measuring compressibility. We demonstrate that, despite the conservative nature of the underlying hyperelastic system, the Stroh operator $\cL$ is non-self-adjoint with respect to the natural weighted inner product. This non-self-adjointness stems from the indefinite metric of the mixed-variable Stroh formulation. We construct the adjoint eigenproblem, identify the corresponding adjoint mode, and utilise the Fredholm solvability condition to derive an explicit first-order formula for the sensitivity of the critical twist with respect to compressibility. Numerical results for a Mooney--Rivlin cylinder reveal a remarkably high sensitivity coefficient ($\kappa_1 \approx 40.3$), indicating that even slight compressibility exerts a strong stabilizing effect on the cylinder. The asymptotic prediction is shown to approximate the full compressible eigenvalue with an error of order $\cO(\varepsilon^2)$. Crucially, we highlight that neglecting the adjoint mode yields erroneous sensitivity predictions, underscoring the necessity of the proposed formulation. Mathematics Subject Classification (2020) 74B20 · 74G60 · 74K10