Analytical and Experimental Study of Beam Torsional Stiffness With Large Axial Elongation

Abstract

The axial force and effective torsional stiffness versus axial elongation are investigated analytically and experimentally for a beam of circular cross section and made of an incompressible material that can sustain large elastic deformation. An approach based on a strain energy function identical to that used in linear elasticity, except with its strain components replaced by those of some finite-deformation tensor, would be expected to provide only limited predictive capability for this large-strain problem. Indeed, such an approach based on Green strain components (commonly referred to as the geometrically nonlinear theory of elasticity) incorrectly predicts a change in volume and predicts the wrong trend regarding the experimentally determined axial force and effective torsional stiffness. On the other hand, use of the same strain energy function, only with the Hencky logarithmic strain components, correctly predicts constant volume and provides excellent agreement with experimental data for lateral contraction, tensile force, and torsional stiffness—even when the axial elongation is large. For strain measures other than Hencky, the strain energy function must be modified to consistently account for large strains. For comparison, theoretical curves derived from a modified Green strain energy function are added. This approach provides results identical to those of the Neo-Hookean formulation for incompressible materials yielding fair agreement with the experimental results for coupled tension and torsion. An alternative approach, proposed in the present paper and based on a modified Almansi strain energy function, provides very good agreement with experimental data and is somewhat easier to manage than the Hencky strain energy approach.