Nonuniform bending theory of hyperelastic beams in finite elasticity

Abstract

This paper deals with the equilibrium problem of slender beams inflexed under variable curvature in the framework of fully nonlinear elasticity. For the specific case of uniform flexion, the authors have recently proposed a mathematical model. In that analysis, the complete three-dimensional kinematics of the beam is taken into account and both deformations and displacements are considered large. In the present paper, the kinematics of the aforementioned model has been reformulated taking into account beams under variable curvature. Subsequently, focusing on the local determination of the curvature, new equilibrium conditions on cross sections are introduced in the mathematical formulation. The governing equations take the form of a coupled system of three equations in integral form, which is solved numerically through an iterative procedure. Therefore, for the generic class of hyperelastic and isotropic materials, explicit formulae for the displacement field, the stretches and stresses in every point of the beam, following both Lagrangian and Eulerian descriptions, are derived. The analysis allows studying a very wide class of equilibrium problems for nonlinear beams under different restraint conditions and subject to generic external load systems. By way of example, the Euler beam has been considered and the formulae obtained have been specialized for a specific neoprene rubber material, the constitutive constants of which have been determined experimentally. The shapes assumed by the beam as the load multiplier increases are shown through some graphs. The distributions of stretches and Cauchy stresses are plotted for the most stressed cross section. Some comparisons are made using a FE code. In addition, the accuracy of the obtained solution is estimated by evaluating a posteriori that the equilibrium equations are locally satisfied.