On finite elements for geometrically-exact planar beams with hyperelastic material models

Abstract

A geometrically-exact planar beam finite-element framework is developed for the analysis of beams with hyperelastic constitutive models. A conventional three-field beam model is first formulated with full geometrical nonlinearity and a linear elastic material law. The kinematics are then enriched by an additional global field that permits deformation through the in-plane thickness. The out-of-plane direction is incorporated separately through plane-strain and plane-stress constitutive reductions, allowing the cross-sectional deformation assumptions and constitutive behaviour to be formulated independently. Numerical results for the present benchmarks show that the plane-stress condition gives essentially coincident centreline deformations for the Poisson's ratio values considered. In contrast, the plane-strain condition, which is widely employed in the literature, makes the formulation sensitive to Poisson's ratio coupling, with the beam becoming significantly stiffer at higher Poisson's ratios. Comparisons between Saint Venant-Kirchhoff and Neo-Hookean hyperelastic models further demonstrate close agreement in the global deformation and in-plane thickness response, with small constitutive differences becoming more apparent in the local out-of-plane stretch. The proposed approach offers a simple and consistent framework to integrate hyperelastic material models into planar geometrically-exact beam models.