Asymptotic Analysis of Torsional Buckling in Nearly Incompressible Hyperelastic Cylinders

Abstract

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 typically sensitive to small deviations from perfect incompressibility. In this work, we investigate the torsional stability of a solid circular cylinder composed of a compressible Mooney–Rivlin material and develop an asymptotic expansion for the critical twist in the nearly incompressible limit. Starting from the exact finite deformation associated with uniform torsion and fixed axial length, we formulate the incremental boundary-value problem in the current configuration. The incremental equations are cast in Stroh form for a six-dimensional state vector collecting displacements and tractions. A rigorous derivation of the incremental constitutive law based on the direct linearization of nominal stress is presented, ensuring consistency in the handling of pre-stress effects. To ensure dimensional consistency and physical clarity, we introduce a systematic non-dimensionalization of the governing equations. We demonstrate that the resulting dimensionless Stroh operator is non-self-adjoint. Constructing the adjoint eigenproblem and utilising the Fredholm solvability condition, we derive an explicit first-order formula for the sensitivity of the critical twist. This formula accounts for both the explicit constitutive relaxation and the implicit redistribution of the base state. Numerical results for a Mooney–Rivlin cylinder reveal a sensitivity coefficient of approximately 5.29, indicating that slight compressibility exerts a strong net stabilising effect. Crucially, the analysis decomposes this effect into two competing mechanisms: a destabilising explicit constitutive relaxation (\(\kappa _{\mathrm{expl}}^{(1)} \approx -1.26\)) and a strongly stabilising relief of the hydrostatic pre-stress (\(\kappa _{\mathrm{impl}}^{(1)} \approx +6.55\)). Furthermore, we uncover a fundamental divergence between standard invariant theories and the volumetric-deviatoric split formulations typically employed in commercial finite element codes. The apparent destabilisation trend often observed in numerical simulations is shown to be an artifact of the decoupled strain energy artificially suppressing the physical pre-stress relief. The results underscore the necessity of incorporating the adjoint mode and carefully evaluating invariant definitions to correctly predict the stability threshold in the nearly incompressible regime.